Liquid Crystalline Blue Phases and Swimmer Hydrodynamics

Liquid Crystalline Blue Phases

and

Swimmer Hydrodynamics

Gareth Paul Alexander

Keble College

Department of Physics, The Rudolf Peierls Centre for Theoretical PhysicsUniversity of Oxford

Thesis submitted in partial fulfilment of the requirements for the degree of Doctor of

Philosophy in the University of Oxford

· Trinity Term, 2008 ·

Liquid Crystalline Blue Phases

and

Swimmer Hydrodynamics

Gareth Paul Alexander

Keble College

Department of Physics, Rudolf Peierls Centre for Theoretical Physics

University of Oxford

Thesis submitted in partial fulfilment of the requirements for the degree of Doctor of

Philosophy in the University of Oxford

· Trinity Term, 2008 ·

Abstract

Soft condensed matter is concerned with the wide variety of materials, ranging from poly-mers to foams, that are neither crystalline, nor gaseous, but lie somewhere in between. Inthis thesis, we shall describe two such “soft” systems: liquid crystalline blue phases and thehydrodynamics of microscopic swimming creatures.

Blue phases epitomise the term liquid crystal, possessing three-dimensional crystalline ori-entational order whilst maintaining full fluidity. We describe some of their unique propertiesand the subtle balance between local order and topology that is responsible for their stability.The application of an electric field leads to both a distortion of the blue phase and to texturaltransitions, both of which we simulate numerically in good agreement with experiments, re-producing the blue phase X transition observed in the 1980s, but not identified theoretically.Finally, we show that blue phases extend beyond their traditional chiral setting and also occurin achiral nematics under strong flexoelectric coupling.

At the small length scales relevant to microscopic bacteria viscous stresses dominates overthe effects of inertia and propulsion is achieved by quite different means to macroscopic organ-isms. We describe in detail the hydrodynamics of a simple model linked-sphere swimmer, withan emphasis on the interactions between swimmers. Time reversal symmetry is important,constraining both the locomotion of individuals and the nature of their interactions. We takethis up with two examples: the hydrodynamic scattering of two swimmers, related to eachother by time reversal, is shown to itself be time reversal covariant; the initial configurationbeing recovered exactly in the final state. Finally, a model of simple dumb-bells is described.Each swimmer is time reversal invariant and does not swim, but collective locomotion is stillpossible through hydrodynamic interactions, leading to a minimal description of active apolarfluids.

i

Contents

Acknowledgements iv

Publications v

1 Introduction to Liquid Crystals 1

1.1 Elastic distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Isotropic-nematic transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Blue phases: a pictorial introduction . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Blue phases: the isotropic–cholesteric transition 13

2.1 Relating the Landau-de Gennes and Frank theories . . . . . . . . . . . . . . . . 142.2 Isotropic–cholesteric transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Blue phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Free energy of the blue phases . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Lattice constants and the redshift . . . . . . . . . . . . . . . . . . . . . 25

2.4 Numerical minimisation: the cholesteric phase diagram . . . . . . . . . . . . . . 252.4.1 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Real space structure of the blue phases . . . . . . . . . . . . . . . . . . . . . . . 342.5.1 Disclination lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.2 Double twist cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Cubic blue phases in electric fields 38

3.1 Electrostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Field induced unwinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Textural transitions: blue phase X . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Direct simulation of blue phase X . . . . . . . . . . . . . . . . . . . . . . 49

4 Flexoelectric blue phases 54

4.1 Flexoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Landau-de Gennes theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Flexoelectric blue phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 Experimental estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4 Flexoelectric effects in the cholesteric blue phases . . . . . . . . . . . . . . . . . 63

4.4.1 Flexoelectro-optic effect in cholesterics . . . . . . . . . . . . . . . . . . . 634.4.2 Blue phase I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.3 Blue phase II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

ii

5 Locomotion: low Reynolds number swimmers 70

5.1 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1.1 Pooley’s corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Hydrodynamics of linked sphere model swimmers 78

6.1 The Najafi-Golestanian swimmer . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Hydrodynamics of a single swimmer . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2.1 Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2.2 Time averaged flow field . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3 Swimming with friends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.3.1 Interaction forces: expanding stokeslets . . . . . . . . . . . . . . . . . . 896.3.2 Swimmer rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.3.3 Swimmer advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.3.4 An improved near field calculation . . . . . . . . . . . . . . . . . . . . . 95

6.4 Trajectories of two parallel swimmers . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Swimmer scattering 102

7.1 Geometry of scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2 Scattering trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.3 T-duality and swimmer scattering . . . . . . . . . . . . . . . . . . . . . . . . . 1047.4 Scattering of the Najafi-Golestanian swimmer . . . . . . . . . . . . . . . . . . . 1087.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8 Dumb-bell swimmers 113

8.1 Hydrodynamic interactions between oscillating dumb-bells . . . . . . . . . . . . 1148.2 Two dumb-bells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.3 Linear chains of dumb-bells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.4 Lattice pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.5 Suspensions of dumb-bells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9 Conclusion 123

References 128

iii

Acknowledgements

I would like to express my sincere gratitude to my supervisor, Prof. Julia M. Yeomans, for herguidance, support and encouragement these past four years. This thesis seems a small returnfor what I have gained.

I also express my thanks to Dr. Davide Marenduzzo and Dr. Christopher M. Pooley, withwhom I collaborated on many aspects of this thesis. Their endless generosity, both intellectualand personal, has been greatly appreciated.

Soft matter at Oxford has grown substantially during my time and I have benefited fromthe arrival of Dr. Ard A. Louis, Dr. Jonathan P. K. Doye, Dr. Dirk G. A. L. Aarts and Dr.Roel P. A. Dullens as new faculty. Julia’s group has also had the privilege of two long termvisitors this last year, Prof. Anna C. Balazs and many of her group from Pittsburgh, andDr. Olivier Pierre-Louis. Their friendship, advice and influx of new ideas have been greatlyappreciated.

I thoroughly enjoyed the opportunity to assist in the organisation of two successful confer-ences held here at Oxford, from which I benefited greatly. The image of Prof. David Queregracefully steering his punt sideways down the Cherwell will live long in the memory.

I have been fortunate to spend my time in Theoretical Physics with an excellent group ofgraduate students, Adam Gamsa, Andrew James, Halim Kusumaatmaja, Mathias Rufino andTimothy Saunders, with whom it has been a joy to work.

It has been a pleasure to share Office 2.4 with so many kind and interesting people. I havegreatly enjoyed the company and stimulus provided by Jen Ryder, Nidhal Sulaiman, KalliFurtado, Halim Kusumaatmaja, Matthew Blow and Vic Putz, from all of whom I have learnta great deal.

Finally, I would like to thank my family, my parents Barbara and Colin and my brotherNeal and his wife Tina, for their continual love and support.

iv

Publications

Part of the work presented in this thesis has appeared in the following peer reviewed publica-tions:

• G. P. Alexander and J. M. Yeomans, “Stabilizing the blue phases”, Phys. Rev. E 74,061706 (2006).

• G. P. Alexander and J. M. Yeomans, “Flexoelectric blue phases”, Phys. Rev. Lett. 99,067801 (2007).

• C. M. Pooley, G. P. Alexander, and J. M. Yeomans, “Hydrodynamic interactions betweentwo swimmers at low Reynolds number”, Phys. Rev. Lett. 99, 228103 (2007).

• G. P. Alexander and D. Marenduzzo, “Cubic blue phases in electric fields”, Europhys.Lett. 81, 66004 (2008).

• G. P. Alexander and J. M. Yeomans, “Dumb-bell swimmers”, Europhys. Lett. 83, 34006(2008).

Further work contributing to this thesis has been submitted for publication:

• G. P. Alexander, C. M. Pooley, and J. M. Yeomans, “Scattering of low Reynolds numberswimmers”.

• I. Dierking, E. Lark, J. Healey, G. P. Alexander, and J. M. Yeomans, “Anisotropy in theannihilation dynamics of topological defects in nematic liquid crystals”.

v

Chapter 1

Introduction to Liquid Crystals

Liquid crystals occupy an intermediate position in the traditional classification of materials

as solids, liquids and gases. They possess characteristics of both fluids and solids, simul-

taneously exhibiting flow like a liquid and elasticity like a solid [1–5]. Typical liquid crys-

talline materials such as 4-pentyl-4′-cyanobiphenyl (5CB), p-azoxyanisole (PAA) and N-(p-

methoxybenzylidene)-p′-butylaniline (MBBA) are composed of long, thin, rod-like molecules.

At high temperatures, or low densities, both the positions and orientations of the molecules

are randomly distributed, with only short range correlations between them. At very low tem-

peratures, or high densities, the molecules arrange themselves in an orderly fashion, occupying

the sites of a lattice and all oriented along a common preferred direction. Liquid crystals exist

in an intermediate regime in which the molecules adopt a preferred direction, but have no long

range correlations in their positions. In continuum theories of liquid crystals this orientational

order is described by a vector field n called the director, which represents a coarse grained

average of the local molecular alignment [1–5].

When polarised light passes through a cell of liquid crystal it emerges unaltered only if the

axis of polarisation is everywhere either parallel or perpendicular to the director. Otherwise

the axis of polarisation will rotate so as to preferentially follow the local orientation of the

liquid crystal. If the cell is placed between crossed polarisers then light will, or will not, pass

through depending on the orientation of the liquid crystal within the cell. This is the principle

behind the beautiful Schlieren textures [1, 6] (Fig. 1.1), the light and dark regions reflecting

the underlying orientational order of the liquid crystal. A typical Schlieren texture displays a

network of points connected by dark threads, immersed in a bright background. Two distinct

types of points may be discerned: those connecting two brushes and those connecting four.

1

1.1 Elastic distortions 2

Figure 1.1: An exemplary Schlieren texture in a nematic liquid crystal. Points with two and fourbrushes correspond to disclinations with topological strength s = 1/2 and s = 1, respectively.Reproduced with the kind permission of Ingo Dierking.

The points themselves correspond to singularities, or defects, in the orientation field where

the director is undefined. The local orientational order in the vicinity of these two types of

singularity is quite different and reveals a very important feature of liquid crystalline order.

Namely, the orientational order in a local neighbourhood encircling a two-brush singular point

is unorientable. Thus the director is apolar and should be described by a line field, instead of

a vector field, as is the case for a magnet.

However, to avoid undue mathematical complexity, and following the usual conventions in

the literature [1–5], we shall continue to treat the director as a vector field and impose, by hand,

the equivalence relation n ∼ −n coming from its apolar character. Just as the orientation of

the director has no physical significance neither does its magnitude convey any information.

Because of this, the director is always normalised to unit magnitude, |n| = 1.

1.1 Elastic distortions

The orientational order in liquid crystals represents the breaking of a continuous symmetry.

As a consequence, global rotations of the director do not cost any energy, but local rotations,

leading to gradients in the director field, do [3]. There are three independent distortions of the

director field in a nematic liquid crystal, known as splay, twist and bend [1–5] and shown in

Fig. 1.2. In terms of derivatives of the director field these three distortions correspond to the

1.1 Elastic distortions 3

Figure 1.2: Schematic illustration of the elastic distortions in a nematic liquid crystal.

following expressions being non-zero:

n(

∇ · n)

splay , (1.1)

n · ∇ × n twist , (1.2)

−(

n · ∇)

n bend . (1.3)

We adopt a slightly unconventional notation here, writing splay as n(∇ · n) instead of simply

∇ ·n and bend as −(n · ∇)n instead of n×∇×n as is more common in the literature [1]. The

latter change is merely a matter of personal preference since the two expressions are equivalent

because the director is a unit vector. We choose to write splay as n(∇ · n) both to emphasise

the requirement that properties of the liquid crystal need to be invariant under n → −n and

to reveal the vectorial character of splay distortions, an aspect of some importance when we

discuss flexoelectricity in Chapter 4.

The preferred state of a liquid crystal is one of uniform alignment and therefore we can

associate a free energy cost with each of these distortions, leading to the Frank free energy

density [1–5]

FFrank = 12K1

(

∇ · n)2

+ 12K2

(

n · ∇ × n + q0)2

+ 12K3

(

(n · ∇)n)2, (1.4)

where K1,K2,K3 are the Frank splay, twist and bend elastic constants, respectively. The

constant q0 appearing in the twist invariant is permitted on symmetry grounds only in systems

lacking inversion symmetry, i.e., in chiral liquid crystals. In such chiral systems this coefficient

will be generically non-zero and serves to promote a naturally distorted texture in which the

director displays a pure twist distortion with n · ∇ × n = −q0. It may readily be verified that

the director field n = (0, cos(q0x), sin(q0x)) satisfies this relation. It describes a helical texture

1.2 Disclinations 4

Figure 1.3: Schematic illustration of topological defects in nematic liquid crystals. (a) s = 1/2,(b) s = −1/2, (c) s = 1, splay-type (ψ = 0), and (d) s = 1, bend-type (ψ = π/2). In each casethe director field is confined to the plane of the page and the location of the disclination lineis indicated by a small black dot.

known as the cholesteric phase in which the director rotates moving along the helical axis, but

is uniform perpendicular to it. The director undergoes a complete revolution over a distance

of 2π/q0, but because of its apolar character the physical repeat length, known as the pitch, is

precisely half this, p = π/q0 [1].

1.2 Disclinations

Topological defects in liquid crystals give rise to both the beautiful Schlieren textures of Fig. 1.1

and the name nematic, which originates from the Greek νηµα meaning ‘thread-like’. They are

singular lines where the director field is undefined, and are classified by a half-integer s, which

represents the winding number, or topological strength, of the disclination line. A schematic

illustration of the local director field configuration for a number of different strength disclina-

tions is shown in Fig. 1.3. The s = ±1/2 ((a) and (b)) and s = ±1 ((c) and (d)) disclinations

have a fundamentally different character. Namely, only the former are topologically protected:

the s = ±1 disclinations can be continuously deformed into a uniform texture and removed by

purely local manipulations. By contrast, no such operations can be performed on the s = ±1/2

disclinations, which can only be removed by the annihilation of a pair of such defects of oppo-

1.2 Disclinations 5

site strength. These remarks, although simple, have important consequences for the switching

dynamics of liquid crystals that we shall describe in Chapters 3 and 4.

A feature of some importance in the description of nematic liquid crystals is that the

Frank theory assigns infinite energy to these disclination lines. In the one elastic constant

approximation, K1 = K2 = K3 = K, the director field in the vicinity of a strength s disclination

takes the form n = (cos(sφ+ ψ), sin(sφ+ ψ), 0), where φ is the usual polar angle and ψ is an

arbitrary constant. The energy of this field configuration is πKs2 ln(R/rc) per unit length of

the disclination line, where rc is a short distance cut-off representing the size of the defect core,

typically of molecular length scales, and R is a large scale cut-off of order the distance to the

nearest boundary or another disclination [1].

The infinite energy cost of s = ±1 disclinations can be avoided by allowing the director to

‘escape in the third dimension’, becoming perpendicular to the page in Fig. 1.3 at the centre

of the defect [7], a situation which is observed experimentally in liquid crystals confined to

cylindrical capillaries [1]. If the boundary conditions are homogeneous at the surface of the

capillary, n = eφ at ρ = R, and escaped along the axis, n = ez at ρ = 0, then the director field

is given by n = cos(θ)ez+sin(θ)eφ, with θ = 2arctan(ρ/R), and the energy of the configuration

is 3πK per unit length of the cylinder. This configuration, generated by a compromise between

geometry and boundary conditions, displays a local twist of the director field as one moves along

any straight line passing through the cylinder axis. For this reason it is reminiscent of the double

twist cylinders that characterise the cholesteric blue phases and which shall be the principal

focus of Chapters 2 - 4. However, the blue phases differ in a fundamental respect: there is no

capillary tube anchoring the liquid crystal at its surface. Thus in the blue phases one needs

both a different mechanism by which to stabilise the locally double twisted configuration, and

to consider how the local texture extends to the surrounding liquid crystal in the region beyond

ρ = R. It turns out that it is not possible to do the latter without introducing disclinations in

the director field as we shall describe in § 1.5.

Finally, we mention that exactly the same comments can be made for a capillary with

homeotropic anchoring, n = eρ at ρ = R, leading to a locally double splayed cylinder. The

prospect that this local configuration might also appear in an unconfined liquid crystal has

been considered only relatively recently [8, 9] and will be taken up in Chapter 4.

1.3 Isotropic-nematic transition 6

1.3 Isotropic-nematic transition

An important consequence of the nature of the local symmetry of liquid crystals is that the

director, being apolar and of fixed (unit) magnitude, cannot serve as an order parameter

for the isotropic-nematic transition. When a nematic liquid crystal is cooled through the

isotropic-nematic transition and viewed through crossed polarisers the sample remains dark

in the isotropic phase, but becomes bright following the transition to the nematic. Thus the

optical response of the liquid crystal can be taken as an order parameter for the transition. In

a continuous medium the optical response is determined by the dielctric tensor ǫ. This tensor

may be split into two pieces, an isotropic part and a deviatoric part, just as can be done for

the stress tensor in a Newtonian fluid,

ǫαβ = 13 tr

(

ǫ)

δαβ +[

12

(

ǫαβ + ǫβα)

− 13 tr

(

ǫ)

δαβ

]

. (1.5)

The isotropic part is present even in the high temperature phase and does not impart any

rotation to the axis of polarisation of light passing through the material. Thus the transmission

of light through crossed polarisers arises solely from the non-vanishing of the deviatoric part of

the dielectric tensor, which we adopt as our order parameter for the isotropic-nematic transition

and denote by the symbol Q. As is usual in Landau theories, the physical origin of the order

parameter is of less direct relevance than its symmetry properties. For example, we could

just as easily have defined our order parameter in terms of the deviatoric part of the magnetic

susceptibility tensor χ [1], and although Q would then have a different physical origin, it would

still have the same symmetries.

For many of our purposes it is therefore only important to note that Q is a traceless,

symmetric, second rank tensor. A coordinate system can always be found for such a tensor in

which it is diagonal

Q = diag(

S,−(S + η)/2,−(S − η)/2)

. (1.6)

When η = 0 the tensor exhibits uniaxial symmetry, which is often assumed to be the case

in nematic liquid crystals. More generally, for η 6= 0, the tensor is biaxial, with η measur-

ing the degree of biaxiality. This biaxiality, which can be important in the description of

disclinations [10] or chiral systems [11], is not captured by the Frank director theory of liquid

crystals.

1.4 Hydrodynamics 7

A Landau theory of the isotropic-nematic transition was first given by de Gennes [1]. The

bulk free energy density is expanded in powers of the independent invariants of the order

parameter, which we take to be tr(Q2) and tr(Q3). The simplest expression for the bulk free

energy density is then

Fbulk = a tr(

Q2)

− b tr(

Q3)

+ c(

tr(

Q2)

)2. (1.7)

The existence of a cubic term in the free energy density indicates that the transition will be first

order, as observed experimentally. The first order nature of the transition raises the question

of whether it is appropriate to truncate the expansion at fourth order, as done here, or also

retain terms up to sixth order. In a mean field calculation, one finds that the fourth order

truncation always yields a first order transition with a uniaxial Q in the nematic phase. If,

however, terms up to sixth order are retained, then not only can the transition lead to a biaxial

nematic, but also there can be second order transitions between two distinct uniaxial nematics

in the ordered phase [1].

1.4 Hydrodynamics

One of the most fascinating aspects of liquid crystals is that they flow. And while the motions of

Newtonian fluids are already extremely rich and complex [12–14], the delicate coupling between

fluid flow and orientational order produces a wealth of new phenomena in liquid crystals [1,15].

In general, flow leads to a rotation of the local orientation, which in turn influences the flow.

Similarly, if a disturbance is initiated in the director, its reorientation is generally accompanied

by fluid motion, an effect sometimes referred to as backflow.

The hydrodynamics of liquid crystals was developed during the 1960s, principally in the

work of Ericksen and Leslie [15], although there was considerable debate over the precise form

of the equations for some time [1, 16, 17]. This formulation, known as the Ericksen-Leslie

equations, is based on the dynamics of the nematic director n and is the most commonly

used description of liquid crystal hydrodynamics. Only the orientational order is treated as a

hydrodynamic variable, the magnitude of the order being considered fixed and homogeneous.

In systems containing defects, or close to a phase transition, however, the magnitude is

not constant and can play a significant role in the hydrodynamics [18–23]. We account for

this by using a hydrodynamic description based on the Q-tensor developed by Edwards, Beris

1.5 Blue phases: a pictorial introduction 8

and Grmela [24–28] and later extended into a general framework for the description of the

hydrodynamics of complex fluids by Grmela and Ottinger [29–31].

The order parameter evolves according to a Ginzburg-Landau type equation describing

relaxation towards the minimum of the free energy, but with a convective time derivative to

account for the advection with the fluid

DtQ = Γ

(−δFδQ

+ 13 tr

(

δF

δQ

)

I

)

. (1.8)

The term in brackets on the right hand side is called the molecular field, H, and Γ is a collective

rotational diffusion constant. The material derivative for rod-like molecules is given by

DtQ =(

∂t + u · ∇)

Q−(

ξD + Ω) (

Q + 13I

)

−(

Q + 13I

) (

ξD− Ω)

+ 2ξ(

Q + 13I

)

tr(

QW)

,

(1.9)

where D = (W + WT )/2 and Ω = (W −WT )/2 are the symmetric and antisymmetric parts,

respectively, of the velocity gradient tensor Wαβ = ∇βuα. The constant ξ depends on the

molecular details of a given liquid crystal.

The fluid velocity field is taken to obey the continuity equation and a Navier-Stokes equation

with a stress tensor generalised to describe liquid crystal hydrodynamics

∂t+ ∇ ·(

u)

= 0 , (1.10)

(

∂tu + u · ∇u)

= −∇p+ ∇ · σ , (1.11)

σαβ = µ(

∇αuβ + ∇βuα)

+ 2ξ(

Qαβ + 13δαβ

)

QγδHγδ

− ξHαγ

(

Qγβ + 13δγβ

)

− ξ(

Qαγ + 13δαγ

)

Hγβ

+QαγHγβ −HαγQγβ −∇αQγδδF

δ∇βQγδ.

(1.12)

Eqs. (1.8) - (1.12) are collectively referred to as the Beris-Edwards equations and we shall

use them throughout this thesis to describe the time evolution of the Q-tensor. To solve these

equations we employ a standard finite difference scheme for the evolution of Q, Eq. (1.8), and

a three-dimensional lattice Boltzmann algorithm [32–34] for the fluid, Eqs. (1.10) and (1.11).


Figure 1.4: (a) Typical texture of blue phase I under reflecting microscopy taken from Ref-erence [50]. (b) A single monodomain of blue phase I exhibiting facetting and growth in asequence of steps, from Reference [35].

1.5 Blue phases: a pictorial introduction

Blue phases are found in highly chiral liquid crystals in a narrow temperature window (typically

< 1 K) between the high temperature isotropic fluid and low temperature cholesteric phases.

They are remarkable mesophases, exhibiting a brightly coloured texture of individual, micron

sized platelets, as shown in Fig. 1.4. The bright colour indicates selective reflection due to

a periodic structure, much like in an ordinary crystal, but with a much larger characteristic

length scale, and indeed the reflection spectra show Bragg peaks that can be indexed by cubic

space groups with lattice constants of several hundred nanometers. Furthermore, individual

platelets of monodomain crystals themselves show distinctive facetting corresponding to Miller

planes of the lattice structure and with the faces growing in a sequence of steps [35]. And yet,

blue phases are not crystals in the traditional sense: they have no long range positional order

and are full three-dimensional fluids [11]. The crystalline order is in the orientational degrees

of freedom of the liquid crystal.

The key to understanding their unique properties was in realising that the locally preferred

ordering in a chiral liquid crystal is one of double twist as opposed to the usual single twist of

the ordinary cholesteric helix [11,36–38]. This local texture of the blue phases can be thought of

as pieces of cholesteric helix simply added together. As an example, consider superposing three

pieces of cholesteric with their helical axes rotated from one another by 120o. As illustrated in

Fig. 1.5 these three pieces combine to produce a local double twist, but in addition, because

of the way they were chosen, they also impose hexagonal symmetry. This hexagonal region


Figure 1.5: A pictorial construction of the planar hexagonal blue phase. Pieces of threecholesteric helices, with their axes rotated from one another by 120o are superposed to forma region of local double twist, which may then be used as a unit cell for a two dimensionalhexagonal lattice. But is the correct unit cell (a) or (b)?

may then be adopted as a unit cell and used to generate a two-dimensional lattice. However,

it is clear that this is not an innocuous process that smoothly extends the local order globally.

Rather, the vertices of the unit cell boundary generate a lattice of s = −1/2 disclination lines,

so that the texture is in this sense frustrated.

The construction we have just described encapsulates the generic features of crystalline

blue phases: they are composed of an array of energetically favourable double twist cylinders

separated by a network of disclinations. This structure also explains why the blue phases are

stable only in a narrow temperature window at the isotropic-cholesteric phase boundary, as

here the magnitude of the order is small and hence the free energy penalty associated with

disclinations is low.

Three distinct blue phase textures have been identified experimentally, separated by first

order phase transitions and occuring in the order blue phase III, blue phase II, blue phase I

upon cooling from the isotropic liquid [39, 40]. Exemplary phase diagrams obtained in Refer-

ence [40] are shown in Fig. 1.6. The textures that are observed experimentally, however, do

not possess the two-dimensional hexagonal symmetry of our simple example. The disclinations

form textures with cubic symmetry and the stable phases have space groups O8−(I4132) and

O2(P4232), corresponding to blue phase I and blue phase II, respectively [41–43]. Blue phase


Figure 1.6: Experimental phase diagrams of the cholesteric blue phases for two different chiralcompounds, taken from Reference [40]. Three distinct blue phases are found, in the order bluephase I, blue phase II, blue phase III upon increasing the amount of chiral dopant.

III is somewhat different and has an amorphous structure with the same symmetry as the

isotropic fluid [11]. The director field texture and properties of blue phase III are not as well

understood as those of the crystalline blue phases [44, 45], which will be the primary focus of

our attention.

The unique combination of crystalline order with lattice constants comparable to the wave-

length of visible light and full three-dimensional fluidity make the blue phases ideal for tech-

nological uses such as fast light modulators, photonic crystals or tunable lasers [46–49]. The

principal obstruction to fulfilling this potential is the very limited temperature range over which

blue phases have traditionally been thermodynamically stable. However, in recent years, novel

techniques have extended the stability range to as much as 60 K, including room temperature,

by the addition of bimesogenic molecules or photo-crosslinking of polymers [50,51]. The devel-

opment of blue phase materials for technological uses has now progressed to the stage where,

in May 2008, Samsung Electronics unveiled a prototype ‘blue phase mode LCD’ at the Society

for Information Display 2008 International Symposium [52].

Many of the potential technological applications of blue phases, like traditional liquid crystal

displays, rely upon the response of the material to an electric field for their operation. In the

blue phases the effects of an electric field are particularly rich, including continuous distortions

of the size and shape of the unit cell and a series of field induced textural transitions to new

blue phase structures, not stable in zero field [11, 53]. This is a classic problem, worked on

extensively during the hey-day of the blue phases in the 1980s, and yet many features remain

only partially explained.


The increased stability of the new blue phase materials is not understood. Traditional de-

scriptions of the blue phases ascribe their stability to a fine energetic balance between favourable

regions of double twist and unfavourable disclination lines. In director field, or low chirality,

theories the latter are thought of as having isotropic cores, which become increasingly costly

as the temperature is lowered below the isotropic transition. To increase their stability one

must either make the double twist regions more beneficial, or the disclinations less costly. It

is natural to speculate that in the case of the polymer stabilised blue phases [51], the polymer

sits preferentially in the defect core regions, whereupon photo-crosslinking effectively solidifies

the texture, but this has not yet been verified.

In the opening part of this thesis we shall investigate the properties of the crystalline blue

phases, presenting first a description of their structure and thermodynamic stability obtained

within the Landau-de Gennes framework, and subsequently an analysis of their behaviour under

an applied electric field.

We close this chapter by recalling the words of Sir Charles Frank when describing blue phases [11].

“They are totally useless, I think, except for one important intellectual use, that of

providing tangible examples of topological oddities, and so helping to bring topology

into the public domain of science, from being the private preserve of a few abstract

mathematicians and particle theorists.”

Chapter 2

Blue phases: the isotropic–cholesteric transition

The theoretical description of the general features and properties of the blue phases was devel-

oped in the 1980s [11,36,37,42,43]. Two approaches were followed: a low chirality theory based

on the Frank director field description [36,37] and a high chirality theory using the Landau-de

Gennes Q-tensor [42,43]. Although broad agreement was reached that the thermodynamically

stable phases all had the cubic symmetry observed experimentally, the agreement with experi-

ment remained incomplete. In particular, the O8− texture identified with blue phase I was not

found to undergo a direct transition to the isotropic liquid, but always appeared between blue

phase II and the cholesteric phase. Furthermore, additional textures with space groups O8+

and O5 were found to be stable within the parameter range corresponding to the experiments,

but were not observed experimentally. Both of these discrepancies were overcome by a numer-

ical minimisation of the Landau-de Gennes free energy [54], within the one elastic constant

approximation.

However, the report by Coles and Pivnenko [50] of a blue phase with a substantially larger

temperature range does not fit into the established theoretical framework and merits closer

attention to understand where its increased stability originates from. In this Chapter we shall

describe a numerical determination of the phase diagram of cholesteric liquid crystals, obtained

for an extended Landau-de Gennes theory, and investigate the effects of varying material pa-

rameters on the blue phase stability. These include the three Frank elastic constant and an

additional chiral invariant that allows for an inversion of the helical sense in the cholesteric

phase. We begin with a review of the relation between the Frank director (§ 1.1) and Landau-

de Gennes (§ 1.3) theories, thereby establishing the extended Landau-de Gennes free energy,

Eq. (2.21), that we minimise to construct the phase diagram.

13

2.1 Relating the Landau-de Gennes and Frank theories 14

2.1 Relating the Landau-de Gennes and Frank theories

The Frank director and Landau-de Gennes Q-tensor theories provide two different ways of

describing the physical properties of liquid crystals. Each of these approaches has a different

emphasis and one often complements the other. Indeed, additional insight can frequently be

gained by comparing and relating the two descriptions.

The director field n may be used to define a traceless, symmetric, second rank tensor

Quniaxial = 3S2

(

n⊗ n− 13I

)

, (2.1)

through which a relationship between the Landau-de Gennes and Frank theories may be estab-

lished. The free parameter S in this mapping represents the magnitude of the liquid crystalline

order and is often referred to as a scalar order parameter. For a given director field, Eq. (2.1)

allows us to construct a Q-tensor. However, by construction this Q-tensor is uniaxial so that,

in general, the procedure cannot be inverted to extract a director field from Q. To do this we

instead note that the director is an eigenvector of Quniaxial that, provided S is positive, corre-

sponds to the maximal eigenvalue. We will continue to use this as a definition of the director

even when Q is not uniaxial. By this definition the director is well-defined and regular so long

as the maximal eigenvalue of Q is unique. Conversely, if the maximal eigenvalue is degenerate

then the director is undefined, indicating the presence of a disclination. Since two eigenvalues

are equal, these disclinations in the director always coincide with points where the Q-tensor is

uniaxial, a necessary and sufficient condition for which is that [55]

∣

∣Q∣

∣

6= 54

(

det(Q))2, and det(Q) < 0 . (2.2)

The Frank free energy, Eq. (1.4), describes the energy penalty associated with spatial

inhomogeneity in the direction of alignment of the liquid crystal. Within the Landau-de Gennes

theory this is accounted for by supplementing the bulk free energy density, Eq. (1.7), with a

gradient free energy density

Fgradient = L1

2

(

∇Q)2

+ L2

2

(

∇ · Q)2

+ 2q0L1Q · ∇ × Q , (2.3)

containing all the independent symmetry allowed invariants up to quadratic order in both


gradients and Q [1]. In the cholesteric systems that will be the focus of our attention it is

frequently convenient to express the gradient free energy density differently, combining the

term linear in gradients with the (∇Q)2 term by ‘completing the square’ [11]

Fgradient = L21

2

(

∇× Q + 2q0Q)2

+ L22

2

(

∇ · Q)2. (2.4)

The two expressions, Eqs. (2.3) and (2.4), are identical up to total derivatives and a tr (Q2)

term, which may be thought of as ‘borrowed’ from the bulk free energy density. The insight

gained from this rewriting of the gradient free energy density is that minimisers can be found

by the solution of a first order differential equation

Q = −(1/2q0)∇× Q , (2.5)

instead of the usual second order Euler-Lagrange equations. The solution of Eq. (2.5),

Q ∼

0 0 0

0 cos(2q0x) sin(2q0x)

0 sin(2q0x) − cos(2q0x)

, (2.6)

is known as a biaxial helix. Here we have chosen the helical axis to lie along the x-direction, but

this is arbitrary and any SO(3) rotation of this tensor will also solve Eq. (2.5). Furthermore,

since Eq. (2.5) is linear, any linear combination of biaxial helices (with any directions for their

helical axes) is also a solution. This is the basis for the pictorial construction, described in

§ 1.5, whereby blue phases are built up as a linear combination of cholesteric helices.

Using Eq. (2.1) it is straightforward to show that the Landau-de Gennes free energy density

reproduces the Frank version, with

K1 = 9S2

4

(

L21 + L22

)

, K2 = 9S2

2

(

L21

)

, K1 = 9S2

4

(

L21 + L22

)

. (2.7)

This highlights a shortcoming of the quadratic Landau-de Gennes theory, which is seen to

account for only two elastic constants, as compared to the three that appear in the Frank

director field theory. To obtain three distinct Frank elastic constants it is necessary to extend

the Landau-de Gennes theory by including terms of cubic order in Q in the gradient free

energy [56–58]. If we continue to confine our attention to terms of at most quadratic order in


gradients then we can construct the following invariants

ǫαβγQαδQδǫ∇βQγǫ , (2.8)

ǫαβγQαδQβǫ∇δQγǫ , (2.9)

Qαβ∇αQβγ∇δQγδ , (2.10)

Qαβ∇αQγδ∇βQγδ , (2.11)

Qαβ∇αQγδ∇γQβδ , (2.12)

Qαβ∇γQαβ∇δQγδ , (2.13)

Qαβ∇γQαγ∇δQβδ , (2.14)

Qαβ∇γQαδ∇γQβδ , (2.15)

Qαβ∇γQαδ∇δQβγ . (2.16)

The mapping of these additional invariants to the Frank theory can be done using Eq. (2.1),

leading to the relations [58]

K1 = 9S2

4

(

L21 + L22

)

+ 9S3

8

(

−L33 − 2L34 − L35 + 2L37 + L38 + 2L39

)

, (2.17)

K2 = 9S2

2

(

L21

)

+ 9S3

8

(

−2L34 + L38

)

, (2.18)

K3 = 9S2

4

(

L21 + L22

)

+ 9S3

8

(

2L33 + 4L34 + 2L35 − L37 + L38 − L39

)

, (2.19)

K2qF0 = 9S2

2

(

L21q0)

+ 9S3

8

(

L31q0 + L32q0)

. (2.20)

If all of the cubic order invariants are retained there will be many more Landau-de Gennes

elastic constants than there are Frank constants and the theory will contain a large number of

phenomenological parameters that it is not realistic to study the effects of in detail. Conse-

quently we shall restrict our attention to a subset, choosing to focus on three invariants; (2.8),

(2.11) and (2.15). These are chosen for a number of reasons: first, it is clear that one should

retain at least one chiral and one achiral invariant. Since there does not appear to be any

substantial difference between the two chiral invariants we feel confident that no qualitative

changes would result from retaining both terms. It is more difficult to choose representatives of

the achiral invariants. The invariant (2.15) is retained because, uniquely amongst the achiral

invariants, it contributes equally to all three Frank elastic constants. This leaves only a choice

of invariant that will distinguish between the Frank splay and bend elastic constants. We

2.2 Isotropic–cholesteric transition 17

choose to use (2.11) partly because it has been used in previous work [18,59], partly because it

gives the largest distinction between K1 and K3 and partly because we found that it has the

largest contribution to the energetics of a single double twist cylinder. The extended version

of the Landau-de Gennes free energy then takes the form

F = 1V

∫

Ωd3r

A0(1−γ/3)2 tr

(

Q2)

− A0γ3 tr

(

Q3)

+ A0γ4

(

tr(

Q2)

)2

+ L21

2

(

∇× Q + 2q0Q)2

+ L22

2

(

∇ ·Q)2

+ q0L31Q2 · ∇ × Q

+ L34

2 Qαβ∇αQγδ∇βQγδ + L38

2 Qαβ∇γQαδ∇γQβδ

,

(2.21)

where in the bulk free energy A0 is a constant with the dimensions of an energy density and γ

plays the role of an effective temperature for thermotropic liquid crystals [28,32].

2.2 Isotropic–cholesteric transition

The description of the transition from an isotropic fluid to a chiral liquid crystal is simplified

by the use of a dimensionless form of the free energy [11, 42], obtained by defining the di-

mensionless length scale x = 2q0r and energy scale f = (729/16A0γ)F , and also rescaling the

order parameter by Q → (4/3√

6)Q. In terms of these dimensionless variables the free energy,

Eq. (2.21), then takes the form

f = 1(2q0)3V

∫

Ωd3x

τ4 tr

(

Q2)

−√

6 tr(

Q3)

+(

tr(

Q2)

)2

+ κ2

4

[

(

∇× Q + Q)2

+ L22

L21

(

∇ ·Q)2

+ 4L31

3√

6L21

Q2 · ∇ × Q

+ 4L34

3√

6L21

Qαβ∇αQγδ∇βQγδ + 4L38

3√

6L21

Qαβ∇γQαδ∇γQβδ

]

,

(2.22)

and is seen to depend only on the two dimensionless parameters

τ :=9(3 − γ)

γ, (2.23)

κ2 :=108q20L21

A0γ, (2.24)

known as the reduced temperature and the chirality respectively, and the ratio of elastic con-

stants.

The isotropic-cholesteric transition can be calculated in detail analytically [11, 42], which


we review briefly, emphasising the additional features arising from the extended form of the

Landau-de Gennes free energy that we use here. The Q-tensor describing an ideal cholesteric

helix is given by [11,42,60]

Q =Q2√

2

0 0 0

0 cos(kx) sin(kx)

0 sin(kx) − cos(kx)

− Qh√6

2 0 0

0 −1 0

0 0 −1

, (2.25)

where we have taken the helical axis to define the x-direction of a Cartesian coordinate system.

A short calculation shows that the free energy is

f = 14(τ+κ2)(Q2

h+Q22)+(Q3

h−3QhQ22)+(Q2

h+Q22)

2+ κ2

4 Q22

[

(1−βQh)k2−2(1−αQh)k]

, (2.26)

where we have defined the parameters α := −2L31/(9L21) and β := 2(2L34 − L38)/(9L21).

Minimisation of Eq. (2.26) with respect to the variational parameters Qh and Q2 leads to the

set of Euler-Lagrange equations

k = 1−αQh

1−βQh, (2.27)

12 (τ + κ2)Qh + 3(Q2

h −Q22) + 4Qh(Q

2h +Q2

2) + κ2

4 Q22

[

−βk2 + 2αk]

= 0 , (2.28)

12(τ + κ2)Q2 − 6QhQ2 + 4Q2(Q

2h +Q2

2) + κ2

2 Q2

[

(1 − βQh)k2 − 2(1 − αQh)k

]

= 0 . (2.29)

Using these, and assuming Q2 6= 0 as expected for a first order transition, we obtain the pair

of equations

Q22 = 1

8

(

12Qh + κ2 (1−αQh)2

1−βQh− (τ + κ2)

)

−Q2h , (2.30)

18Q2h + κ2Qh

(1−αQh)2

1−βQh= Q2

2

(

6 − κ2

2

[

2α− β 1−αQh

1−βQh

]

1−αQh

1−βQh

)

, (2.31)

which formally define the equilibrium Q-tensor. In general these equations must be solved

numerically; however, they can be easily solved when α = β = 0, and we recover the usual

results [11,42,61]

Qh = 9−κ2

48

(

1 +√

1 − 72τ(9−κ2)2

)

, (2.32)

Q22 = 27+κ2

24 Qh − 3τ32 . (2.33)


The main feature which arises from the inclusion of higher order terms in the gradient free

energy is that the helical wavevector now depends on the temperature. As Eq. (2.27) shows,

the wavevector depends on the amplitude of the homogeneous part of the order parameter, and

since the order parameter depends on the temperature, it follows that so does the cholesteric

pitch. It should be noted that, even with such a simple modification of the free energy, we

can already account for the inversion of the helical sense with decreasing temperature observed

experimentally in some cholesterics [1, 62,63]. According to the present theory the cholesteric

wavevector changes sign when Qh = 1/α. When this value is substituted into Eqs. (2.30)

and (2.31) we obtain the following expression for the inversion temperature

τHI = 4α2 (3α− 8) − κ2 . (2.34)

At the inversion itself we find Q2 =√

3Qh and

Q =2

α√

6

−1 0 0

0 2 0

0 0 −1

, (2.35)

which corresponds, as expected, to a uniaxial nematic with director n = (0, 1, 0). Other

orientations of the nematic within the yz-plane may be obtained by adding a constant phase

shift to the sin and cos terms in Q in Eq. (2.25). It is clear that within this theory, as

observed experimentally, the helical inversion is a smooth cross-over that does not involve any

discontinuities in physical quantities, i.e., it is not a phase transition.

The isotropic-cholesteric transition temperature is obtained by supplementing Eqs. (2.30)

and (2.31) with the condition that the free energy (Eq. (2.26)) be zero. A short calculation

leads to the following equation for Qh at the transition temperature:

(

4Qh + κ2

4(1−αQh)2

1−βQh

)(

6 − κ2

21−αQh

1−βQh

[

2α− β 1−αQh

1−βQh

]

)2

−(

24Qh + κ2(1−αQh)2

1−βQh− κ2Qh

21−αQh

1−βQh

[

2α− β 1−αQh

1−βQh

]

)2

= 0 . (2.36)


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ

Isot

ropi

c tr

ansi

tion

tem

pera

ture (i)

(ii)

(iii)

(iv)

(a)

−6 −5 −4 −3 −2 −1 0 1−1.5

−1

−0.5

0

0.5

1

τ

Cho

lest

eric

wav

evec

tor

(i)

(ii)

(iii)

(iv)

(b)

Figure 2.1: (a) Isotropic cholesteric transition temperature as a function of chirality for a rangeof parameter values: (i) α = β = 0, (ii) α = 2 , β = 0, (iii) α = 2 , β = 1, (iv) α = 4 , β = 0.(b) Temperature dependence of the cholesteric wavevector for the same set of parameters as in(a) (all cases are for κ = 0.05).

Again when α = β = 0 we recover the usual results [11,42,61]

QICh = 1

8

(

1 − κ2

3 +

√

1 + κ2

3

)

, (2.37)

τIC = 12

(

1 − κ2 +(

1 + κ2

3

)3/2)

, (2.38)

where τIC is the temperature at which the isotropic-cholesteric transition occurs. A situation

of some experimental interest occurs when the cholesteric undergoes helix inversion at the

isotropic-cholesteric transition temperature [62]. This case may be solved exactly by setting

Qh = 1/α in Eq. (2.36). We find α = 4 and

τHI = τIC = 1 − κ2 , (2.39)

which is precisely the transition temperature for a nematic (with non-zero κ). This particular

case provides a guide to the range of values that α can be expected to take in experimental

systems.

In the general case Eq. (2.36) can be solved numerically. Fig. 2.1(a) shows the isotropic-

cholesteric transition temperature as a function of chirality for a selection of values of α and

β. The general trend can be understood on the basis of the effect that the parameters have on

the helical pitch. Increasing α increases the pitch and shifts the liquid crystal towards nematic

behaviour, while increasing β decreases the pitch and hence shifts away from the nematic state.

2.3 Blue phases 21

The temperature dependence of the helical wavevector is shown for the same set of values of

α and β in Fig. 2.1(b). It can be seen that the wavevector varies approximately linearly with

temperature except in the immediate vicinity of the transition temperature. It may at first seem

unusual that increasing β should lead to a larger temperature dependence, however this can be

understood by noting that the inversion temperature depends only on α and that increasing β

leads to a larger wavevector at the isotropic transition temperature.

Finally, it is known that if the chirality is sufficiently large then the isotropic-cholesteric

transition is second order instead of first [11, 42]. We can calculate the maximum value of κ

compatible with a first order transition by setting Qh = 0 in Eq. (2.36). The resulting equation

is solved to obtain

κmax = 22α−β

[

−1 +√

1 + 3(2α − β)]

. (2.40)

In what follows it will be seen that this limit is never reached due to the intervention of blue

phases.

2.3 Blue phases

The platelet-like visual appearance and Bragg scattering of blue phases I and II suggest that

they may be well approximated by a periodic Q-tensor. It is therefore convenient to introduce

a Fourier decomposition [42,60]

Q(r) =∑

k

N−1/2k

2∑

m=−2

Qm(k) eiψm(k) Mm(k) e−ik.r . (2.41)

Here Qm, ψm are the amplitude and phase of the Fourier mode, Nk is a convenient normalising

factor equal to the number of Fourier modes k that all have the same magnitude k, and the

tensors Mm represent a set of basis tensors adapted to each Fourier mode. Reality of the

Q-tensor imposes the constraint

e−iψm(k)M†m(k) = eiψm(−k)Mm(−k) , (2.42)

which we choose to satisfy by requiring separately that ψm(−k) = −ψm(k) and M†m(k) =

Mm(−k). The basis tensors Mm(k) should themselves be chosen so as to diagonalise the

quadratic part of the Landau-de Gennes free energy. This calculation was first done by Bra-

2.3 Blue phases 22

zovskii and Dmitriev [60] who showed that the appropriate choice is

M±2 =1

2

(

v ± iw)

⊗(

v ± iw)

,

M±1 =1

2

[

(

v ± iw)

⊗ k + k⊗(

v ± iw)

]

,

M0 =1√6

(

3k ⊗ k − I)

.

(2.43)

Here the vectors v,w are two unit vectors chosen such that the trio k,v,w form a right-

handed, orthonormal set. However, this choice is not unique, in the sense that the vectors

v,w may be freely rotated about the direction k and still form a right-handed, orthonormal

set. Thus there is a U(1) gauge freedom in the choice of the basis tensors. Under a gauge

transformation, where v,w are rotated through an angle Θ in the right-handed sense, the basis

tensors transform as Mm → eimΘMm.

The Q-tensor for a periodic blue phase is obtained by specifying the set of wavevectors

to appear in the Fourier decomposition and the values for the phases ψm(k), given for a

particular choice of gauge for the tensors Mm(k). The choice of lattice type determines the

set of wavevectors, while the phases ψm(k) are determined by the particular space group of

the candidate blue phase. An application of Buerger’s method [64] shows that the symmetry

operations of the space group lead to the relations

ψm(Rk) = ψm(k) + k.t − (k − Rk).y +mΘ(k,Rk) mod 2π , (2.44)

between the phases of the Fourier modes. Here R is the rotational part of the symmetry

operation, t is the translational component along the rotation axis, y is the point on the

rotation axis closest to the origin, and Θ(k,Rk) is an additional phase angle coming from the

gauge transformation between RMm(k)RT and Mm(Rk).

Eq. (2.44) places considerable restrictions on the possible space groups that a blue phase

texture can have. We illustrate this for the most immediately relevant case of cubic textures,

although the results also apply to tetragonal textures, which we discuss in Chapter 3. All cubic

systems possess four-fold rotational symmetry about the cubic axes: we consider the effect of

a four-fold rotation about the z-axis on the wavevectors (2π/Lz)lez. In this case Rk = k and

2.3 Blue phases 23

the gauge angle Θ(k,Rk) is simply π/2, so that Eq. (2.44) reduces to

2π lp/4 +mπ/2 = 0 mod 2π , (2.45)

with p denoting the order of the screw axis. For energetic reasons the m = 2 modes are the

most important ones to consider [42, 60]. Restricting our attention to this case we find that

if the lattice is simple cubic, l = 1, then the four-fold axis has p = 2, while if the lattice is

body-centred cubic, l = 2, then the four-fold axis has p = 1. Thus generically, simple cubic

textures have 42 screw axes and body-centred cubic textures have 41 screw axes. An exception

is possible where the amplitude Q2(k) vanishes, which in fact occurs for the O5 texture [42].

We do not show explicitly the remainder of the calculation of the Fourier phases, but only

record the results. For blue phase I the space group is O8−(I4132) and the phases are [42,43]

ψ2(110) = ∓π ; ψ2(110) = ±π/2 , plus cyclic permutations , (2.46)

where we have adopted the conventional shorthand (hkl) notation for reciprocal lattice vectors.

For blue phase II the space group is O2(P4232), for which the phases are [42,43]

ψ2(100) = 0 , plus cyclic permutations . (2.47)

In both cases the phases are given relative to a natural gauge in which the local basis vectors

v,w are chosen to lie along the lattice directions, e.g., for k = ex we choose v = ey and

w = ez. It is noteworthy that two choices are possible in the case of blue phase I: in fact

these correspond to distinct blue phases, each possessing different disclination networks [11].

We shall follow the convention adopted in the literature and refer to the texture arising from

choosing the upper sign as O8− and that from choosing the lower sign as O8+.

2.3.1 Free energy of the blue phases

The Fourier decomposition, Eq. (2.41), provides a means to calculate analytically the free

energy of a blue phase texture. When this form of the Q-tensor is inserted into the free energy,

2.3 Blue phases 24

Eq. (2.22), we obtain [42,43]

f =∑

k

N−1k

∑

m

Q2m(k)

τ+κ2

4 + κ2

4

[(

1 + 4−m2

6L22−L21

L21

)

k2 −mk]

−√

6∑

k1,k2,k3

N−1/2k1

N−1/2k2

N−1/2k3

∑

m1,m2,m3

Qm1(k1)Qm2

(k2)Qm3(k3)

× ei(ψm1(k1)+ψm2

(k2)+ψm3(k3)) δk1+k2+k3

[

1 + κ2

42L38

9L21

(

k2 · k3

)

]

tr(

Mm1(k1)Mm2

(k2)Mm3(k3)

)

+ κ2

42L34

L21δm2,m3

k2 · Mm1(k1) · k3 − ακ2

4 tr(

ik3 × Mm1(k1)Mm2

(k2)Mm3(k3)

)

+∑

k1,k2,k3,k4

N−1/2k1

N−1/2k2

N−1/2k3

N−1/2k4

∑

m1,m2,m3,m4

Qm1(k1)Qm2

(k2)Qm3(k3)Qm4

(k4)

× ei(ψm1(k1)+ψm2

(k2)+ψm3(k3)+ψm4

(k4)) δk1+k2+k3+k4

× tr(

Mm1(k1)Mm2

(k2))

tr(

Mm3(k3)Mm4

(k4))

.

(2.48)

The large number of terms contributing to the cubic and quartic invariants makes an explicit

calculation of this free energy for any chosen blue phase texture a very laborious process. On

the small number of occasions when this has been seriously attempted [11,42,43,57,61,65] the

Fourier sum has been truncated to a small number of wavevectors and to m = 2 modes only, in

order to keep the calculation manageable. The restriction to m = 2 modes is motivated from

a consideration of the quadratic free energy, which is minimised by the choice m = 2. This

is often justified by considering the high chirality limit, κ → ∞, however, the conclusion is

only arrived at if the gradient free energy is restricted to quadratic order in Q. Furthermore,

if only m = 2 modes are retained in the Fourier expansion then the free energy of the blue

phase is independent of the elastic constant L22 so that the quadratic gradient theory shows no

sensitivity to the relative magnitude of the Frank splay, twist and bend elastic constants. Since

these are observed experimentally to greatly influence the blue phase stability [66] we must

conclude that m 6= 2 modes contribute significantly to the blue phase free energy. Nonetheless,

the restriction to m = 2 modes provides an appropriate first approximation, both for analytic

and numerical calculations, and we shall use it to construct Q-tensors that serve as the initial

condition for numerical simulations.

2.4 Numerical minimisation: the cholesteric phase diagram 25

2.3.2 Lattice constants and the redshift

The pitch of the cholesteric helix is determined by the strength of chiral dopant. In both the

Frank director and Landau-de Gennes theories this is encoded by the constant q0 appearing

in the free energy density. Similarly, the lattice constants of blue phases I and II are also

determined by the strength of the chiral dopant, q0, however, they need not coincide with the

value of the cholesteric pitch. Experimentally it is found that the blue phase lattice constants

are larger than the helical pitch, typically by 10-20%. This observation is accommodated

naturally within the Landau-de Gennes theory by minimising Eq. (2.48) with respect to the

magnitude of the primitive reciprocal lattice constant, 2π/a. The wavevectors in a cubic system

have magnitude (2π/a)(h2 + k2 + l2)1/2, so that the reciprocal lattice constant is given by [42]

2π

a=

∑

〈h,k,l〉∑

mQ2m(h, k, l) (h2 + k2 + l2)1/2 (m/2)

∑

〈h,k,l〉∑

mQ2m(h, k, l) (h2 + k2 + l2)

, (2.49)

if we restrict to the quadratic gradient theory. For the cholesteric texture, the Q-tensor con-

tains a single Fourier wavevector, (100) say, corresponding to the mode m = 2, so that in

dimensionless units we have 2π/p = 1 (Eq. (2.27) with α, β = 0). Re-instating the dimensions

we recover the familiar result p = π/q0 [1].

In the infinite chirality limit, the Q-tensors describing crystalline blue phases contain only

the fundamental set of wavevectors and we find that 2π/a = (h2 + k2 + l2)−1/2 so that the

magnitude of these wavevectors is again k = 2q0 [11,42], as in the cholesteric phase. However,

for any finite value of the chirality, the presence of higher harmonics will lead to the magnitude

of the fundamental wavevector taking a value lower than 2q0. This decrease can be quantified

by a parameter called the redshift, which is defined through the relation k =: 2q0 r between

the magnitude of the fundamental wavevector and the Landau-de Gennes chiral parameter q0.

2.4 Numerical minimisation: the cholesteric phase diagram

We now describe the phase diagram resulting from a minimisation of the modified Landau-de

Gennes free energy, Eq. (2.22), taking into account the two cubic blue phases. Some progress

in obtaining an analytic description of the blue phases is possible, see, e.g., Reference [65],

however, this approach involves adopting an approximate form for the order parameter and

only gives a constrained minimisation. Furthermore, althougth the analytic theory correctly


identifies the two structures observed in experiments, it does not reproduce the correct order of

appearance of the two phases at low chiralities. A full minimisation can be achieved numerically

and has been described in Reference [54] for the one elastic constant approximation.

To study the different phases, cholesteric, blue phase I or blue phase II, it is necessary to

implement appropriate initial conditions for the simulation. The Q-tensor is initialised using

analytic expressions appropriate to the high chirality limit, which act to define the symmetry

of the chosen phase. These are constructed following the Fourier analysis of § 2.3, truncating

the Fourier series at the fundamental set of wavevectors and restricting to m = 2 modes only.

For blue phase I we use [11,42]

Qxx ∼ − sin(ky/√

2) cos(kx/√

2) − sin(kx/√

2) cos(kz/√

2)

+ 2 sin(kz/√

2) cos(ky/√

2) ,

Qxy ∼ −√

2 sin(kx/√

2) sin(kz/√

2) −√

2 cos(ky/√

2) cos(kz/√

2)

+ sin(kx/√

2) cos(ky/√

2) ,

(2.50)

with the other components obtained by cyclic permutation. Similarly, for blue phase II the

Q-tensor is initialised as [11,42]

Qxx ∼ cos(kz) − cos(ky) ,

Qxy ∼ sin(kz) ,

(2.51)

with the other components again obtained by cyclic permutation. We comment briefly that

these expressions are not identical with those given in the seminal work of Grebel, Hornreich

and Shtrikman [42]. This is due to a minor error in that work, which the expressions quoted

here correct.

Under subsequent numerical evolution according to Eq. (1.8) the system relaxes to that

structure of the same symmetry which minimises the free energy. We are therefore able to

obtain, for any value of the parameters, local minima of the free energy corresponding to each

of the cholesteric and blue phases. The global free energy minimum was taken to be the smallest

of these calculated local minima.

We have seen in § 2.2 that the inclusion of cubic invariants in the gradient free energy leads to

a temperature dependent helical pitch in the cholesteric phase, Eq. (2.27). For the blue phases

as well, the unit cell size is temperature dependent, so that to achieve a full minimisation of the


free energy it is necessary to set the correct unit cell size in the simulation. This unit cell size

is not known a priori, but rather depends on the magnitude of the order parameter, a quantity

that is only determined by the numerical minimisation. Therefore we must introduce a means

of determining, and setting, the unit cell size as the Q-tensor evolves during the simulation.

We can account for a change in unit cell size by rescaling the gradient contributions to the free

energy and molecular field. This is accomplished in practice by changing the elastic constants

as follows

q0 = qinit0 /r ,

L2a = Linit2a × r2 ,

L3b = Linit3b × r2 ,

(2.52)

where a = 1, 2, b = 1, . . . , 9, a superscript ‘init’ denotes the initial value of a simulation

parameter and r is the appropriate rescaling factor, which is identical to the redshift described

in § 2.3.2. One shortcoming of analytic calculations [42,43] is that the value of the redshift is

not determined exactly, but only for the approximate form of the Q-tensor that is assumed.

Similarly, in previous numerical work [54], the redshift was assumed to take the value suggested

by the approximate analytic calculations. The exact redshift for the cholesteric phase could

be calculated by obtaining the numerical value of Qh and using Eq. (2.27). However, a similar

approach is not available for the blue phases and consequently it is more useful, and easier, to

calculate it using the free energy.

To do this we note that since the free energy is quadratic in gradients, it may be written

formally in k-space as

f = ak2 + bk + c , (2.53)

where the coefficients a, b and c depend on the Q-tensor, but not on k. The optimum wavevector

is given by k = −b/2a, and since the coefficients a and b are determined by the simulation at

every timestep it is straightforward to use these values to determine the exact value for the

redshift. This is not a dynamic process: the redshift is simply set to this optimal value and

does not relax towards equilibrium.

In order to verify that the procedure was working successfully, and to check the level of

accuracy that could be obtained, we used this numerical method to calculate the wavevector

of a cholesteric undergoing helix inversion and compared it to the theory described in § 2.2.


−4 −3 −2 −1 0 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

τ

Cho

lest

eric

wav

evec

tor

(a)

−4 −3 −2 −1 0 1

−4000

−3000

−2000

−1000

0

1000

2000

3000

τ

Cho

lest

eric

pitc

h

(b)

Figure 2.2: Comparison of numerical (⋄) and analytic (solid line) calculations of the helicalpitch in a cholesteric displaying helical sense inversion. (a) Cholesteric wavevector and (b)pitch against reduced temperature, τ .

Simulation parameters were chosen to set α = 2.0 and κ ≈ 0.096, leading to an inversion

temperature of τHI ≈ −2, Eq. (2.34). The results are shown in Fig. 2.2. We have plotted both

the helical wavevector, which is the relevant quantity theoretically, and the pitch, since this is

more frequently given in experimental work. As can be seen the agreement is excellent even

very close to the inversion point.

2.4.1 Phase diagram

The phase diagram for chiral liquid crystals obtained for a selection of parameter values using

the extended Landau-de Gennes free energy, Eq. (2.21), is shown in Fig. 2.3. These were

obtained numerically using a lattice Boltzmann algorithm [32] to solve the Beris-Edwards

equations for the evolution of Q, Eqs. (1.8) - (1.12). A single unit cell was simulated and

periodic boundary conditions imposed in all directions. We show first, in Fig. 2.3(a), the phase

diagram obtained for the one elastic constant approximation with numerical optimisation of

the unit cell size. For comparison, the phase diagram for a fixed unit cell size determined in

Reference [54] is reproduced in Fig. 2.3(b).

Although the qualitative features remain unchanged, we note that optimisation of the unit

cell size has extended the range of stability of blue phase I, both relative to the cholesteric

phase and relative to blue phase II. The movement of the cholesteric phase boundary is quite

significant, with the triple point moving to lower chirality by about 20%. This is due to the

optimum redshift taking a lower value at these chiralities than was assumed previously. Based

on analytic calculations of Grebel et al [42, 43] the redshift was assigned the value 0.79 in


Reference [54]. However, we find a much lower value, with average 0.68 at the cholesteric-blue

phase I phase boundary (the value is roughly independent of temperature, except very close

to the isotropic transition temperature). In contrast, at the blue phase I-blue phase II phase

boundary we obtain an average redshift of 0.77 for blue phase I and 0.86 for blue phase II.

These values are in better agreement with the analytic results of Grebel et al [42,43], primarily

because the chirality at this phase boundary is much closer to the values of chirality where the

analytic calculations predict the blue phases are stable. In what follows we will use Fig. 2.3(a)

as a reference phase diagram relative to which the effect of varying the Landau-de Gennes

parameters can by measured.

We now consider how the phase diagram changes as the ratios of the elastic constants are

varied. Here one expects that the qualitative features of the phase diagram will be retained,

but it is nonetheless of interest to determine how large a quantitative shift can be obtained.

Fig. 2.3 shows the phase diagrams obtained upon separately varying the twist and bend elastic

constants.

To investigate the effect of the bend elastic constant we chose parameter values L21 = L22 =

L34 = 0.02, L38 = 0 which corresponds to a ratio of bend to splay of about 1.75, while splay

and twist remain degenerate. The stability of blue phase I is seen to decrease quite significantly

relative to the cholesteric phase while at the same time there is a small increase in stability

over blue phase II. There is only a minor shift in the cholesteric-blue phase I phase boundary

at the transition temperature, however, as the temperature decreases the shift becomes larger.

For example at a reduced temperature of τ = −2 the phase boundary occurs at a chirality of

κ ≈ 1, representing a shift to higher chiralities of almost 70% as compared to the one elastic

constant case. The value of the redshift is markedly increased, with the average value for

blue phase I at the cholesteric boundary being 0.81. However, it was shown in § 2.2 that the

cholesteric wavevector depends on the value of L34 and hence has also increased. This raw

value of the redshift no longer represents the most directly accessible quantity and a more

relevant figure is the ratio of the blue phase I redshift to that of the cholesteric, since this can

be measured experimentally by means of the discontinuity of the back-scattered Bragg peak.

For this quantity we obtain an average of 0.67 and note the strong similarity of this value with

that obtained in the one elastic constant approximation.

At the blue phase I-blue phase II phase boundary the redshift is 0.87 and 0.97 for blue phase

I and blue phase II respectively. Again, it is the ratio of these values which is more directly


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−4

−3

−2

−1

0

1

κ

τ

Isotropic

Cholesteric

O8−

O2

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−4

−3

−2

−1

0

1

κ

τ

Isotropic

Cholesteric O8−

O2

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−4

−3

−2

−1

0

1

κ

τ

Isotropic

Cholesteric

O8−

O2

(c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−4

−3

−2

−1

0

1

κ

τ

Isotropic

Cholesteric

O8−

O2

(d)

0 0.2 0.4 0.6 0.8 1

−5

−4

−3

−2

−1

0

1

κ

τ

Isotropic

O8−

N*

helix inversion

(e)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.75

0.8

0.85

0.9

0.95

1

κ

τ

Isotropic

O8−

O2

Cholesteric

(f)

Figure 2.3: Numerical evaluation of the phase diagram of cholesteric liquid crystals within theLandau-de Gennes theory. τ , the reduced temperature, and κ, the chirality, are defined byEqs. (2.23) and (2.24). Phase diagrams in the one elastic constant approximation with (a)optimisation of the unit cell size, and (b) fixed unit cell size, after [54]. Phase diagrams withunequal elastic constants: (c) K1 = K2 = 0.5K3, (d) K1 = K3 = 1.5K2. (e) The numericalphase diagram obtained with the chiral invariant, Eq. (2.8), added to the free energy. Themagnitude of this term was chosen so as to produce helical sense inversion in the cholestericphase at a temperature not far below the isotropic transition temperature. (f) An enlargementof the region near the isotropic transition temperature. Note the reversal in the order ofappearance of blue phase I and blue phase II as a function of chirality.


relevant to experiment. The ratio of the O8− unit cell size to that of O2 is 0.90, and again we

note a strong similarity with the ratio obtained from the one elastic constant approximation,

0.89.

The value of the twist elastic constant is controlled by the Landau-de Gennes parameter

L22. In most liquid crystals the twist elastic constant is smaller than either splay or bend. In

order to match this, we constructed the phase diagram for parameter values L21 = 0.02, L22 =

0.04, L34 = L38 = 0, which is shown in Fig. 2.3(d). This choice of parameters resulted in a ratio

of splay to twist of about 1.5, while splay and bend remained degenerate. Again we observe

that the stability of blue phase I is reduced relative to the cholesteric phase by an amount

similar to that seen by varying the bend elastic constant. The values of the redshift for both

blue phase I and blue phase II are only very slightly increased relative to their values in the

one elastic constant limit, while the cholesteric wavevector is insensitive to the value of L22.

This reveals an intriguing feature, that while the phase boundaries and absolute values of the

redshift can vary appreciably, the ratios of the redshift for the different phases, and hence the

discontinuities in back-scattered Bragg peaks, are essentially independent of the values of the

elastic constants.

Finally, we consider the effect of the chiral cubic invariant on the blue phases. We chose

parameter values of L21 = L22 = L34 = 0.02, L38 = 0.04, α = 2.0, which for the cholesteric

phase sets β = 0 and gives a ratio of bend to splay of about 1.6. The phase diagram is shown

in Fig. 2.3(e).

The value of α is such that the cholesteric undergoes helical sense inversion at a reduced

temperature of about τ ≈ −2. What is remarkable is the dramatic increase in stability of

blue phase I relative to the cholesteric phase. The region of stability has been increased down

to chiralities as low as κ = 0.07 and at such low chiralities the phase boundary is essentially

independent of κ for all τ . In addition, we find a very small region of stability for blue phase II

located close to the isotropic transition. As shown in greater detail in Fig. 2.3(f) this occurs in

a narrow temperature interval at chiralities lower than those for which blue phase I is stable,

representing a reversal of the order of appearance of the two blue phases. In the region where

blue phase II is stable we find a redshift of 0.36. In contrast to the situation without the chiral

invariant, this is smaller than the blue phase I redshift, which takes the value 0.43. Again,

the absolute values of the blue phase redshifts are not as relevant as their ratios to that of the

cholesteric phase. In this case we find the cholesteric redshift is 0.49 at the isotropic transition


−5 −4 −3 −2 −1 0 1

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

τ

wav

evec

tor BPI

Cholesteric

(a)

−5 −4 −3 −2 −1 0 1

−40

−30

−20

−10

0

10

20

τ

pitc

h

BPI

Cholesteric

(b)

Figure 2.4: Numerical results for the temperature dependence of the (a) wavevector and (b)pitch of the unit cell of blue phase I along the cholesteric-blue phase I phase boundary atκ ≈ 0.07 (⋄) . For comparison the cholesteric wavevector and pitch are also given: numericalresults () , results from the calculation presented in § 2.2 (solid line).

temperature, giving ratios of 0.88 for blue phase I and 0.74 for blue phase II, both of which are

significantly different to those obtained in the one elastic constant limit.

Since blue phase I is now stable over a much larger temperature range it displays a significant

variation in unit cell size as the temperature is lowered. As an illustration of this, the blue

phase I redshift is 0.19 at a reduced temperature of τ ≈ −5, corresponding to more than a

two-fold increase in the lattice parameter. A plot of the temperature dependence of the blue

phase I redshift is shown in Fig. 2.4. For comparison the helical wavevector of the cholesteric

phase is also plotted on the same graph.

2.4.2 Discussion

We have investigated numerically the phase diagram of the cholesteric blue phases for a range

of parameter values within the framework of an extended Landau-de Gennes theory. The tra-

ditional Landau-de Gennes theory has long been known to only accommodate two independent

Frank elastic constants and to have a temperature independent cholesteric pitch. Both of these

shortcomings were overcome by retaining terms of cubic order in the Q-tensor in the expansion

for the gradient free energy. Since the new terms were added specifically to remove the degen-

eracy between splay and bend, and to give a temperature dependence to the helical pitch they

possess a clear and simple physical interpretation. In particular, the value of the parameter α,

which controls the strength of the chiral cubic invariant, should be relatively easy to estimate

on the basis of Eq. (2.34) for the helix inversion temperature. The magnitude of the elastic


constants for the achiral cubic invariants is more difficult to ascertain. Although this may

be estimated from the ratios of the Frank elastic constants it is clear that, because there are

more Landau-de Gennes elastic constants than Frank, the latter are insufficient to uniquely

determine the former (an estimation of the magnitude of the cubic Landau-de Gennes elastic

constants was given in Reference [56]).

The extended Landau-de Gennes theory that we have investigated provides a phenomeno-

logical description of helical sense inversion in the cholesteric phase. The inversion arises as a

natural consequence of the presence of including higher order chiral invariants in the gradient

free energy. Although such higher order contributions are usually neglected since they are

deemed small compared to the terms already retained, the fact that helix inversion is observed

experimentally demonstrates that these terms can play a significant role. We also note that,

for systems undergoing first order phase transitions, such as liquid crystals, although the order

parameter is small it is not infinitesimal, and therefore the relative magnitude of a given term

in the Landau expansion depends not only on the power of the order parameter but also on

the size of any numerical coefficient premultiplying it.

The primary aim of this Chapter has been to investigate how much the properties of the

blue phases can be changed within the framework of Landau-de Gennes theory. In this regard

we have shown that the retention of cubic order terms in the gradient free energy can lead

to considerable changes in the size of the blue phase unit cell and in their phase diagram.

Most dramatic amongst the results is the increase in stability of blue phase I obtained in

systems where the cholesteric undergoes helical sense inversion. Again we comment that this

significant, qualitative change in the phase diagram arises from retaining cubic order terms and

demonstrates that these give rise to more than just small changes in the physical properties.

It is of interest to consider whether the mechanism considered here is a candidate to account

for the increased range of stability in blue phase I recently reported by Coles and Pivnenko [50].

It seems not, as apart from the large temperature range most features of their blue phase differ

from those obtained for the choice of parameters we made here. For example, the numerics

show an increase in the size of the blue phase I unit cell with decreasing temperature, while in

the experimental system the unit cell size shows a small decrease. Also, numerically we find

that blue phase II has a larger unit cell than blue phase I at the transition between the two,

in contrast to what is found experimentally. However, we remark that the parameter space in

the Landau-de Gennes theory is large enough that these discrepancies could well be resolved

2.5 Real space structure of the blue phases 34

by a different choice of parameters.

2.5 Real space structure of the blue phases

We conclude this Chapter with a brief description of the real space structure and texture of

the cubic blue phases, partly for completeness, partly to emphasise some features of double

twist cylinders not mentioned in the extant literature and partly to explain the meaning of the

figures we will show in Chapters 3 and 4. We focus on blue phase II for simplicity, but also

provide numerical results for blue phase I.

The approximate expressions for the Q-tensor, Eqs. (2.50) and (2.51), contain only the

fundamental set of Fourier wavevectors and are restricted to the m = 2 modes. As such, they do

not lead to accurate estimates of the free energies of the cubic blue phases. In particular, at this

level of truncation, the expression for blue phase II, Eq. (2.51), yields a vanishing cubic invariant

and thus predicts a second order transition from the isotropic phase. Nonetheless, because they

capture the correct symmetries, the simple expressions given in Eqs. (2.50) and (2.51) provide

a remarkably accurate description of the texture of the cubic blue phases. In particular, they

correctly give the location and structure of both the disclination network and the double twist

cylinders; the two most identifying features of the blue phases. We outline this for the case of

blue phase II.

2.5.1 Disclination lines

The location of the disclinations may be identified from the requirement that the maximal

eigenvalue of Q is degenerate so that the director field is undefined. Eq. (2.2) leads to the

condition

(

3 − cos(kx) cos(ky) − cos(ky) cos(kz) − cos(kz) cos(kx))3

= 27 sin2(kx) sin2(ky) sin2(kz) ,

(2.54)

which can be seen by inspection to be satisfied everywhere along the body diagonals, x, y, z =

±(2π/k)t. Along the [111] body diagonal the Q-tensor takes the form

Q ∼ sin(2πt)

0 1 1

1 0 1

1 1 0

, (2.55)


so that in a frame in which Q is diagonal we have

Q ∼ sin(2πt) diag (2,−1,−1) . (2.56)

From this we see that the subleading eigenvalues are degenerate for 0 < t < 1/2 and the

Q-tensor is uniaxial, but there is no defect, while for 1/2 < t < 1 the maximal eigenvalue is

degenerate indicating the presence of a disclination line. The network of these disclination lines

extending along half of the body diagonals is shown in Fig. 2.5.

It is easier to identify the disclinations numerically from the magnitude of the order (more

correctly, of the maximal eigenvalue of Q) than from the criterion that two eigenvalues are

degenerate. Within the defect cores, the large energy cost of the disclination is relieved by

reducing the magnitude of the order, allowing the core region to become partially isotropic.

Thus in all our images of the disclination network the defects are identified by plotting an

isosurface, everywhere within which the magnitude of the order drops below a threshold value.

2.5.2 Double twist cylinders

Unlike the disclination lines, there is no mathematical procedure analogous to Eq. (2.2) to

identify double twist cylinders in a blue phase texture and our only guides are symmetry and

past experience. Here, we propose to identify the axes of the double twist cylinders using the

criterion that the director is parallel to one of the cubic directions everywhere along the cylinder

axis. We shall sketch in outline that this correctly identifies the double twist structure of blue

phase II, and show in Chapter 4 that this also extends to the deformation of the cylinders

under an applied electric field.

We consider in turn the three lines (1/2, 0, t), (t, 0, 1/2) and (1/2, t, 1/2). For the first line

we have

Q(1/2, 0, t) ∼

−1 + cos(2πt) sin(2πt) 0

sin(2πt) −1 + cos(2πt) 0

0 0 2

. (2.57)

The eigenvalues are 0,±2, so that Q is maximally biaxial, and the director is given by n =


(0, 0, 1) ∀ t. Along the second line we find

Q(t, 0, 1/2) ∼

−2 0 0

0 1 + cos(2πt) sin(2πt)

0 sin(2πt) 1 − cos(2πt)

, (2.58)

so that again the eigenvalues are 0,±2, but this time the director is not constant and instead

takes the form n = (0, cos(πt), sin(πt)). Finally, along the third line the Q-tensor is given by

Q(1/2, t, 1/2) ∼

−1 − cos(2πt) 0 sin(2πt)

0 0 0

sin(2πt) 0 1 + cos(2πt)

. (2.59)

The eigenvalues are now 0,±2 cos(πt) and the director, in the region about t = 0, is given by

n = (sin(πt/2), 0, cos(πt/2)).

The structure we have just described is, of course, a double twist cylinder with the line

(1/2, 0, t) describing the cylinder axis. However, unlike the double twist cylinders which form

the basis of the low chirality theories of blue phases, this cylinder is biaxial and not uniaxial.

Consequently its cross-section is elliptical instead of circular. A numerical visualisation of the

double twist cylinders throughout the unit cell, shown in Fig. 2.5, reveals further that this

cross-section is not uniform along its axis, but shows a periodic modulation commensurate

with the lattice. Here we are showing an isosurface everywhere inside which the director has

a projection onto one of the cubic directions that is larger than some threshold value. Finally,

we comment that in the case of blue phase I the cylinder axes are not straight lines, as has

been commonly assumed in the literature.


(a)

(b)

Figure 2.5: Network of disclinations and arrangement of double twist cylinders in blue phaseI (a) and blue phase II (b). The disclinations are shown on the left in blue and the doubletwist cylinders on the right in grey. In all figures 23 unit cells have been shown to more clearlyillustrate the structure.

Chapter 3

Cubic blue phases in electric fields

When an external electric field is applied to a liquid crystal it provides a preferred spatial

direction to which the director field can couple, allowing the orientation of the liquid crystal to

be controlled. In a dielectric medium an electric field contributes an amount (−ǫ0/2)EαǫαβEβto the free energy density. For a uniaxial material, such as a nematic liquid crystal, the dielectric

tensor has different magnitudes parallel and perpendicular to the director and may be written

ǫαβ = ǫ‖nαnβ + ǫ⊥(

δαβ − nαnβ)

. (3.1)

Thus in a uniaxial nematic an applied electric field couples to the director via a term in the

free energy density of the form

Fdielectric =−ǫ0(ǫ‖ − ǫ⊥)

2

(

E · n)2. (3.2)

If the dielectric anisotropy is positive (ǫ‖ > ǫ⊥) then the director will attempt to re-align parallel

to the field, whilst if it is negative the director will prefer to be everywhere perpendicular to

E.

Since the cholesteric blue phases have an orientational order which displays cubic symmetry

with a lattice constant of several hundred nm they have the potential to serve as lasers, fast

light modulators or photonic crystals [46–49]. Moreover, because they are fluids, their structure

can be easily manipulated and hence the properties of blue phase devices are widely tunable.

A natural way of seeking to do this is through the application of an external electric field.

However, because of the intricacy of their director field configurations, the response of the blue

38

39

phases to an applied electric field is more complicated than that of a uniaxial nematic.

Experimental work in the 1980s and 1990s identified the principal features to be electrostric-

tion, a continuous distortion of the shape and size of the unit cell, and a series of field induced

textural transitions [53,67–75]. The electrostriction involves a shift of the back-scattered Bragg

peak of 5 − 10%, and is quadratic in the field strength. The direction of the shift is observed

to change with the sign of the dielectric anisotropy, but blue phase I also displays an unusual

response referred to as anomalous electrostriction, where an expansion is seen when the field

is oriented parallel to the [011] direction, but a contraction for fields parallel to [001].

At larger field strengths new blue phases appear, possessing tetragonal or hexagonal sym-

metry [53, 70–75]. The possibility of a transition to a two-dimensional hexagonal texture was

first proposed theoretically by Hornreich, Kugler and Shtrikman [76], shortly after which field

induced transitions were reported experimentally [70]. Three distinct field induced textures

were identified, possessing tetragonal, screw hexagonal and two-dimensional hexagonal sym-

metry with increasing field strength [53,70–72].

Much of the electric field behaviour has been understood via extensions of the Landau–de

Gennes theory [76–81], providing qualitative agreement for the quadratic field dependence of

the electrostriction and the sensitivity to the sign of the dielectric anisotropy. The qualitative

features of the field induced hexagonal phases, including both the screw and two-dimensional

textures, have also been reproduced [79]. However, the approximations necessary in performing

semi-analytic calculations lead to predictions for the electrostriction coefficients which are an

order of magnitude too small, limiting the quantitative comparison which is possible. At

a qualitative level, the anomalous electrostriction of blue phase I was not found within the

Landau-de Gennes theory [78,80,81] and the tetragonal texture known as blue phase X has not

been included in any previous theoretical calculations [79]. Moreover, all previous studies have

been restricted to statics and it has not been possible to investigate changes in the director

field nor the pathway by which disclinations rearrange during textural transitions.

In this Chapter we describe a numerical investigation of the effects of an electric field

on the cubic blue phases by using a three-dimensional lattice Boltzmann algorithm to solve

the hydrodynamic equations of motion. We present results for the electrostriction in good

agreement with experiment, obtaining, in particular, the correct order of magnitude for the

shift in Bragg peak and the anomalous electrostriction of blue phase I. Furthermore, we are

able to follow the rearrangement of defects and unwinding of the cubic blue phases at large

3.1 Electrostriction 40

field strengths, and, by reproducing the relevant experimental set-up, we can investigate for

the first time the continuous blue phase I–blue phase X transition.

3.1 Electrostriction

The electrostriction, or change in size and shape, of the blue phase unit cell may be described

by a minimal extension of the Landau-de Gennes theory, first introduced by Lubin and Horn-

reich [77]. A dielectric term

Fdielectric = −∆ǫ EαQαβEβ , (3.3)

is added to the usual free energy density and the equilibrium state obtained by minimisation.

In doing this, the high chirality approximation of the Q-tensor is generalised to describe the

non-cubic geometries induced by the symmetry breaking of the electric field. For small field

strengths the distorted unit cell geometry may be described by relating the primitive lattice

vectors under an applied electric field to those in zero field according to

ai(E) =(

δij +RijαβEαEβ + o(E4))

aj(0) , (3.4)

where ai are the set of primitive lattice vectors and Rijαβ are the components of the elec-

trostriction tensor. The absence of a term linear in Eα in Eq. (3.4) can be argued both because

typical experiments are performed using an ac field, for which the time averaged linear term

vanishes, and because there are no non-zero third rank tensors Pijα compatible with the point

symmetry (432) possessed by the cubic blue phases [69].

The approximate, high chirality form of the Q-tensor is taken to be the same as in the

absence of an electric field, but constructed with respect to the distorted lattice and with a

homogeneous component Qh added so that the dielectric term yields a non-zero contribution to

the free energy. The values of the components of the electrostriction tensor are then determined

by treating them as variational parameters in the Landau-de Gennes free energy. Previous

theory [77, 78, 81] has shown that for the cubic blue phases with point symmetry (432) there

are only three independent components of the electrostriction tensor, which we take to be

R1 ≡ Rzzzz, R2 ≡ Ryyzz and R3 ≡ 2Ryzyz. When the field is applied parallel to the [001]


direction the distorted lattice vectors are given by

ax,y(E) =(

1 +R2E2)

ax,y(0) , (3.5)

az(E) =(

1 +R1E2)

az(0) , (3.6)

so that this configuration allows for two of the components of the electrostriction tensor to be

determined. The final component, R3, can be obtained by applying the field parallel to the

[011] direction, for which the lattice vectors become

ax(E) =(

1 +R2E2)

ax(0) , (3.7)

ay,z(E) =(

1 + 12

(

R1 +R2

)

E2)

ay,z(0) + 12 R3E

2az,y(0) . (3.8)

The anomalous electrostriction of blue phase I corresponds to the condition R1/R3 < 0. Exper-

imentally it has been found that, for positive dielectric anisotropy, R3 > 0 for both blue phases,

R1 < 0 and R2 > 0 for blue phase I, while R1 > 0 and R2 < 0 for blue phase II [53]. Typical

magnitudes of the components of the electrostriction tensor are ∼ 1 − 3 × 10−2µm2V −2, with

R1 and R3 approximately equal in magnitude and R2 between 2 and 3 times smaller [53,69].

Analytic calculations using the Landau-de Gennes approach [77–81] have obtained the cor-

rect signs for the components of the electrostriction tensor in blue phase II, but not in blue

phase I, so that, in particular, the anomalous electrostriction was not reproduced. The mag-

nitudes of R1, R2 and R3 are found to be an order of magnitude smaller than those observed

experimentally, although the ratios are in good general agreement.

The principal reason for this discrepancy and the difficulty in performing accurate analytic

calculations is that the electric field couples directly only to the homogeneous component of Q.

Its influence on the blue phase structure is thus indirect, occuring through the coupling between

different order parameter components introduced by the cubic and quartic invariants of the

bulk free energy. Thus to accurately capture the effects of the electric field it is necessary

to accurately calculate the cubic and quartic invariants, which are precisely the terms that

are most difficult to determine analytically. A key advantage of the numerical approach we

adopt here is that it allows for an exact treatment of the effect of these invariants. In previous

numerical work on blue phases in zero field this was found to significantly improve the agreement

between theory and experiment [54,82], as discussed in Chapter 2.


Figure 3.1: Variation of the free energy with varying unit cell shape, blue phase I (left) andblue phase II (right): The free energy is determined for several values of the ratio Lz/Lxparameterising the distortion (markers) and fitted to a quadratic (solid lines). In the leftpanel, the left and right axes correspond to E = 0.0008 and E = 0.0016 respectively. In theright panel, the left and right axes correspond to E = 0.0016 and E = 0.0031 respectively.

3.1.1 Numerical simulation

The response of the cubic blue phases to the applied electric field was determined using a lattice

Boltzmann algorithm [32,34] to solve the Beris-Edwards equations for the combined evolution

of the Q-tensor and the fluid velocity [28], Eq. (1.8) - (1.12). Hydrodynamic effects have

previously been shown to have a significant effect on the dynamics of defects [18,20,21,23] and

on the rheological properties of blue phases [83]. The correct incorporation of hydrodynamics

may therefore be important in determining the switching of blue phases in response to an

applied field, although this is not the focus of our attention in this Chapter.

The cubic blue phases were initialised using the approximate, high chirality expressions for

the Q-tensor, Eqs. (2.50) and (2.51), suitably modified to account for the non-cubic geometry

induced by the electrostriction. A single unit cell was simulated and periodic boundary con-

ditions imposed in all directions. Values of the simulation parameters were chosen to ensure

that we are far from the blue phase-cholesteric phase boundary at zero field. The value of

the chirality was set to κ= 0.7 and κ= 1.86 for blue phase I and blue phase II respectively,

while τ = 0 throughout. This models the higher chirality of blue phase II which arises in

experiments upon increasing the amount of chiral dopant. In the following time and electric

field are referred to in simulation units. These can be related to physical units by using the

conversion that one simulation unit corresponds to 0.01 and 0.0015 µs for blue phase I and blue

phase II respectively, if the rotational viscosity is 1 P. The strength of the electric field can be


inferred from the critical field required to unwind a cholesteric texture, and one simulation unit

then corresponds to ∼350 V µm−1. We restrict our attention to positive dielectric anisotropy,

∆ǫ>0.

A fundamental difficulty in simulating electrostriction is in correctly accounting for the

distortion of the unit cell. Within the Landau-de Gennes theory the distortion is obtained by

the condition that the size and shape of the unit cell are chosen so as to minimise the free

energy, F . We account for this distortion in two steps: first the shape of the simulation unit

cell is fixed and we determine the value of the redshift which minimises the free energy for

this fixed shape of unit cell, using the technique described in § 2.4. We then perform a series

of such simulations varying the shape of the unit cell, e.g., from cubic to tetragonal. Fig. 3.1

shows exemplary results of this procedure for fields applied parallel to the [001] direction of

the blue phase unit cell. The free energy is determined for several values of the ratio Lz/Lx

parameterising the cubic-tetragonal distortion and fitted to a quadratic. The actual distortion

at any free energy is then given by the miminum of this fit. The position of the minimum at

values of Lz/Lx smaller than 1 for blue phase I and larger than 1 for blue phase II is clear

evidence of anomalous electrostriction.

We first consider the case of a uniform field parallel to the [001] direction, with the field

strength varied from zero up to values large enough to induce an unwinding of the blue phase

structure to a uniformly aligned nematic. In order to gain a more complete picture, we have

looked at both the case in which the blue phase unit cell is forced to retain its cubic symmetry

and also when it is allowed to become distorted to tetragonal symmetry, accounted for by simply

varying the size of the unit cell along the ([001]) z-direction. The field dependence of the blue

phase lattice parameters is shown in Fig. 3.2. The response is markedly different depending

on whether or not the shape of the unit cell is allowed to change. When cubic symmetry is

enforced there is very little change in the unit cell size until the field strength becomes large.

Furthermore, the field dependence of this shift is not quadratic even at low field strengths,

although it fits quite well to a quartic. This suppression of electrostriction is a consequence

of the incompatibility of a homogeneous component in an order parameter possessing cubic

symmetry [78]. An expansion is seen for both blue phases, hence R1 > 0 for both blue phases

and the anomalous electrostriction of blue phase I is not reproduced.

When the shape of the unit cell is varied, the magnitude and qualitative features of the

response change considerably. In both blue phase I and blue phase II we observe the correct


0 0.002 0.004 0.006 0.008E (simulation units)

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

Rel

ativ

e un

it ce

ll si

ze

0 0.002 0.004 0.006 0.008E (simulation units)

1

1.01

1.02

1.03

1.04

Rel

ativ

e un

it ce

ll si

ze

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007E (simulation units)

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Rel

ativ

e un

it c

ell s

ize

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007E (simulation units)

0.96

1

1.04

1.08

1.12

1.16

Rel

ativ

e un

it c

ell s

ize

Figure 3.2: Field induced distortion of the unit cell of blue phase I (left) and blue phase II(right): In the top row the shape of the unit cell has been fixed with its initial cubic symmetry,while in the bottom row we have allowed for a distortion of the shape to give tetragonalsymmetry. The lattice parameters parallel (, solid line) and perpendicular (, dashed line)to the field are shown relative to their zero field value.

3.2 Field induced unwinding 45

direction of the shift in lattice parameters, i.e., the correct signs of R1 andR2, [53]. The different

behaviour, a contraction for blue phase I and an expansion for blue phase II, parallel to the

field, is a clear indication of the anomalous electrostriction of blue phase I. This is confirmed

by simulations with blue phase I and the field parallel to the [011] direction, predicting an

expansion of the unit cell along the field direction. Remarkably, anomalous electrostriction has

never been explained within the free energy in Eq. (2.21): our simulations suggest that this

is due to truncation in the Fourier decomposition previously used to calculate electric field

induced deformations. We also observe that the field dependence of the shift is only quadratic

at low field strengths. Such behaviour has not been reported experimentally, however, it should

be noted that our simulations consider only perfect, free floating single crystals and that the

discrepancy may be due to the difficulty of achieving such idealised conditions experimentally.

By fitting our results for the field dependence of the lattice parameters to a quadratic

function at low field strengths we can extract R1 and R2. In simulation units R1 = −18.8 ×

103 , R2 = 9.4 × 103 for blue phase I and R1 = 2.7 × 103 , R2 = −1.0 × 103 for blue phase II.

Mapping these to real units we obtain values in the range 10−2 − 10−1µm2 V −2 for both blue

phases, in good agreement with experiments [53]. If the distortion of the unit cell is volume

preserving, R1/R2 is exactly −2 [78]. We find this to be the case for blue phase I, but not for

blue phase II where instead we obtain R1/R2 ≈ −2.7, in close agreement with the value (−2.6)

experimentally reported in Reference [67]. In order to estimate R3 we also need to consider

the case when the field is applied parallel to the [011] direction and the unit cell undergoes an

orthorhombic distortion. Focusing on simulations at small E we estimate R3 = 19.6 × 103 in

simulation units for blue phase I.

3.2 Field induced unwinding

In the intermediate to large field limit, when the distortion of the size and shape of the unit

cell becomes considerable, it is to be expected that this will be accompanied by a distortion of

the director field and disclination network of the blue phase. As we have already commented,

this is an important effect with particular relevance to field induced phase transitions and the

switching dynamics of potential blue phase devices. Here we look at two scenarios: in the first

we consider the disclination dynamics of blue phase II when switched to the nematic phase

by a field applied parallel to the [001] direction. Then we look at the set-up which gives rise

3.3 Textural transitions: blue phase X 46

experimentally to a continuous blue phase I-blue phase X transition [72].

A large part of the qualitative behaviour may be understood on the basis of simple topolog-

ical considerations. The defect networks in the cubic blue phases are composed of topologically

protected s=−1/2 disclination lines. As such, they can only be unwound if two such disclina-

tions merge into a single s=−1 disclination, since the latter is topologically unstable. As we

shall see, in order to do this, the defects develop a twist, converting themselves from straight

lines into helices.

The switching of blue phase II is shown in Fig. 3.3(I). In these simulations the shape of

the unit cell is fixed as cubic at all times, however the dynamics are unchanged when a (fixed

shape) tetragonal unit cell is used instead. The switching starts with an ‘unbinding’ of the bi-

continuous network to give a layered arrangement of isolated s=−1/2 disclinations, oriented

perpendicular to the field. These isolated disclinations then develop a twist which enables them

to ‘reconnect’ into a new bi-continuous network. Again this ‘unbinds’, this time leaving an array

of pairs of s=−1/2 disclinations oriented parallel to the field and wrapping around each other

in a double helix configuration. Finally, these merge into an unstable s=−1 disclination which

unwinds to leave a defect free nematic texture.

3.3 Textural transitions: blue phase X

Next, we consider the situation in which blue phase I is placed under an intermediate field

applied parallel to the [011] direction. This is the set-up corresponding to the blue phase I-

blue phase X transition observed experimentally in the 1980s [71, 72], and never understood

theoretically. The electrostriction distorts the shape of the unit cell until it becomes tetragonal.

Starting from this geometry, and an unperturbed blue phase I, we numerically follow the

evolution of the disclination network, as shown in Fig. 3.3(II). Initially, the disclinations in

the network twist, they then merge to form a transiently branched structure, which finally

reorganises into a new defect network, not stable at zero field. This is a candidate structure for

blue phase X, as it is a new network, found via a continuous reorganisation starting from blue

phase I, and only stable in a field. Our results also predict that in our candidate blue phase X

(i) the disclinations perpendicular to the field are largely unaffected, (ii) the network conforms

to the space group D104 identified in Reference [72], and (iii) the double twist cylinders deform

but do not break during the transition (contrary to the speculations in Reference [72]; this


(I) (II)

(B)

(C)

0

0

1

2

10

0.5

1

00 1 2

(A)

0

0 1 2 0 1 2

1

2

1 2

0.5

1

00 0.5 1

0.5 1

1

2

01 2

00 1 2 0

0.5

10.5010.50

1

x/Ly/L

z/L

z/L

y/L

x/Ly/L

z/Lz/L

y/Lx/L

x/L

z/L

y/L 0 0.5 10 0.5y/L

z/L

x/L

x/Lx

x

x

x

xx

y

yy

y

yy

zz

zz

zz

Figure 3.3: Switching dynamics of (I) the blue phase II–nematic transition induced at E =0.0094 with the field parallel to the [001] direction (vertical in the figure), and (II) the bluephase I–blue phase X transition with E = 0.006, parallel to the [011] direction (vertical in thefigure). The grey tubes represent regions of the fluid inside which the magnitude of the orderfalls below a threshold value and thus show the network of disclinations. In (I) we have shown23 unit cells for clarity, while only a single unit cell is shown in (II). From top to bottom, times(in simulation units, measured from the application of an electric field) corresponding to thesnapshots are: 0 (A), 7000 (B), 45000 (C) for blue phase II and 0 (A), 80000 (B), 200000 (C)for blue phase I. Axis labels are given in fractions of the unit cell size (Lx, Ly and Lz alongthe x, y and z directions).


0 0.25 0.5 0.750

0.25

0.5

0.75

0 0.25 0.5 0.750

0.25

0.5

0.75

(a) (b)

0.5

(c) (d)0 1

0

1

0.5

0

0.5

1

10 0.5

0 00

z/L z/

L

y/L

x/L

y/L

y/L

x/L

y/L

1

2

0

2

1

1 2 1 2xx

yy

yz

yz

Figure 3.4: Top: director field in blue phase II for fields parallel to [001], shown in the yz-planeat 1/2 lattice parameter distance along the x-axis. (a) E = 0, (b) E=0.008. Bottom: directorfield in blue phase X (E = 0.006) viewed along the tetragonal axis, showing (c) hexagonal arrayof double twist cylinders, and (d) stacking faults. The slices in (c) and (d) are separated by1/4 lattice parameter distance along the z axis. Axis labels are given in fractions of the unitcell size (Lx, Ly and Lz along the x, y and z directions).

observation may be verified by experiments along the lines of those in References [84,85]).

Finally, it is also of interest to look at the static changes that occur within the unit cell.

Some examples of the distortion induced in the director field are shown in Fig. 3.4. In blue phase

II we see a change in the director due to unbinding of the network even for moderate values of the

electric field (Fig. 3.4(a) and (b)). In the gap that opens up between the separated defects the

director is well-aligned parallel to the field, representing the appearance, and growth, of nematic

layers within the blue phase structure. Fig. 3.4(c) and (d) show details of the configuration

of the director field in the blue phase X structure, in two planes perpendicular to the [001]

direction (defects highlighted). We find that the director adopts a ‘screw hexagonal’ texture

where layers of perfect hexagons are separated by layers of ‘stacking faults’. Qualitatively, this


may also explain why larger fields witness the transition to hexagonal blue phases.

3.3.1 Direct simulation of blue phase X

As an additional confirmation that the field induced blue phase I transition we have obtained

numerically is indeed the transition to blue phase X observed experimentally, we would like

to directly initialise the blue phase X texture and confirm that it has the same disclination

network and configuration of double twist cylinders. In order to do this we need to obtain the

appropriate high chirality form of the Q-tensor with which to initialise the simulation, which

can be done by following the general procedure described in § 2.3.

Definitive experiments in the 1980s by Pieranski and Cladis [72] showed that blue phase X

is centred tetragonal and can be described by the space group D104 (I4122). In order to generate

the required body centred tetragonal symmetry, we take the fundamental set of wavevectors

to be

k =

± 2πLx

(

ex + ey)

,± 2πLx

(

−ex + ey)

,± 2πLx

(

ex + Lx

Lzez

)

,

± 2πLx

(

ey + Lx

Lzez

)

,± 2πLx

(

−ex + Lx

Lzez

)

,± 2πLx

(

−ey + Lx

Lzez

)

,

(3.9)

where Lx, Lz are the lattice constants perpendicular and parallel to the tetragonal axis, respec-

tively. For the specification of the the basis tensors Mm(k) we need to assign a choice of gauge

to each of these wavevectors, which we do as follows

k = (110) : v = 1√2

(

−ex + ey)

, w = ez , (3.10)

k = (110) : v = 1√2

(

−ex − ey)

, w = ez , (3.11)

k = (101) : v = ey , w = 1√1+(Lx/Lz)2

(

−(Lx/Lz)ex + ez)

, (3.12)

k = (011) : v = −ex , w = 1√1+(Lx/Lz)2

(

−(Lx/Lz)ey + ez)

, (3.13)

k = (101) : v = −ey , w = 1√1+(Lx/Lz)2

(

(Lx/Lz)ex + ez)

, (3.14)

k = (011) : v = ex , w = 1√1+(Lx/Lz)2

(

(Lx/Lz)ey + ez)

. (3.15)

Finally we need to assign the phase angles ψm(k) in a manner that is consistent with the

symmetries of the space group D104 (I4122). The full list of point symmetries for this space

group can be found in the International Tables [86], however, for our purposes we only need to

consider the four-fold symmetry about the tetragonal axis and the two-fold symmetry about


the y-axis. The former is a 41 screw axis with, in the notation of Eq. (2.44), t = (Lz/4)ez and

y = (−Lx/4)ex+(Lx/4)ey, while the two-fold axis has t = 0 and y = (Lx/4)ex+(3Lz/8)ez [86].

Consider first the four-fold axis: under such a transformation ex → ey, ey → −ex, ez → ez

and for our choice of gauge Θ = 0. Applying Eq. (2.44) to the (110) and (110) wavevectors

leads to the relations

ψm(110) = ψm(110) + π , (3.16)

ψm(110) = ψm(110) − π , (3.17)

from which we immediately deduce that

ψ2(110) = 0, π and ψ2(110) = π, 0 . (3.18)

Similarly, applying Eq. (2.44) in turn to the (101), (011), (101) and (011) wavevectors yields

the relations

ψm(011) = ψm(101) + 3π2 , (3.19)

ψm(101) = ψm(011) + π2 , (3.20)

ψm(011) = ψm(101) − π2 , (3.21)

ψm(101) = ψm(011) + π2 . (3.22)

Combining these we may deduce further that ψm(101) = ψm(101) and ψm(011) = ψm(011).

Finally, by considering the effect of the two-fold axis on the (101) and (011) wavevectors we

obtain the additional relations

ψm(101) = ψm(101) − π2 , (3.23)

ψm(011) = ψm(011) − 3π2 , (3.24)

from which we conclude that

ψ2(101) = ψ2(101) = π4 ,

−3π4 and ψ2(011) = ψ2(011) = −π

4 ,3π4 . (3.25)

It is noteworthy that, as for the space group O8(I4132) relevant to blue phase I, there is


more than one choice of the phase angles ψm(k) compatible with D104 (I4122) symmetry. In

this case there are four distinct possibilities since the two choices in Eq. (3.18) can be made

independently of the two choices in Eq. (3.25).

With a choice of gauge defined and the Fourier phases determined, a straightforward cal-

culation gives the high chirality form of the Q-tensor

Qxx = −eiψ2(110) Q2(110)√4

sin(

2πLx

(x− x0))

sin(

2πLx

(y − y0))

− Q2(101)√8

[

2 cos(

2πLx

(y − y0))

sin(

2πLz

(z − z0) − ψ2(101))

+ 21+(Lz/Lx)2

cos(

2πLx

(x− x0))

cos(

2πLz

(z − z0) − ψ2(101))

]

,

(3.26)

Qyy = −eiψ2(110) Q2(110)√4

sin(

2πLx

(x− x0))

sin(

2πLx

(y − y0))

+ Q2(101)√8

[

2 cos(

2πLx

(x− x0))

cos(

2πLz

(z − z0) − ψ2(101))

+ 21+(Lz/Lx)2

cos(

2πLx

(y − y0))

sin(

2πLz

(z − z0) − ψ2(101))

]

,

(3.27)

Qxy = −eiψ2(110) Q2(110)√4

cos(

2πLx

(x− x0))

cos(

2πLx

(y − y0))

+ Q2(101)√8

2√1+(Lz/Lx)2

[

cos(

2πLx

(y − y0))

cos(

2πLz

(z − z0) − ψ2(101))

− cos(

2πLx

(x− x0))

sin(

2πLz

(z − z0) − ψ2(101))

]

,

(3.28)

Qyz = eiψ2(110) Q2(110)√4

√2 cos

(

2πLx

(x− x0))

sin(

2πLx

(y − y0))

+ Q2(101)√8

2√1+(Lz/Lx)2

[

sin(

2πLx

(x− x0))

cos(

2πLz

(z − z0) − ψ2(101))

− 1√1+(Lx/Lz)2

sin(

2πLx

(y − y0))

cos(

2πLz

(z − z0) − ψ2(101))

]

,

(3.29)

Qxz = −eiψ2(110) Q2(110)√4

√2 sin

(

2πLx

(x− x0))

cos(

2πLx

(y − y0))

+ Q2(101)√8

2√1+(Lz/Lx)2

[

sin(

2πLx

(y − y0))

sin(

2πLz

(z − z0) − ψ2(101))

− 1√1+(Lx/Lz)2

sin(

2πLx

(x− x0))

sin(

2πLz

(z − z0) − ψ2(101))

]

,

(3.30)

where (x0, y0, z0) is the location of the origin within the simulation unit cell. The two choices

for ψ2(110) differ only by the overall sign of the Q2(110) terms. Similarly, the two choices for

ψ2(101) differ only by the overall sign of the Q2(101) terms. However, the difference between

these two choices may equally well be absorbed into a shift of z0, so that we do not expect them

to lead to distinct blue phase textures. Thus we expect two distinct D104 textures corresponding

to the two choices for ψ2(110). By analogy with O8 we denote these as D10±4 , with the plus


sign corresponding to the choice ψ2(110) = π.

This is indeed what is found in simulations and the two resulting textures are shown in

Fig. 3.5. The D10+4 texture is seen to agree with the disclination network of our candidate blue

phase X obtained by switching from blue phase I (Fig. 3.3(II)). Furthermore, in all simulations

it was found to have lower free energy than the D10−4 texture under identical conditions.

Moreover, the D10−4 texture was found to be less numerically robust than D10+

4 . In particular,

the form of the disclination network is sensitive to the relative magnitude of the two coefficients

Q2(110) and Q2(101) appearing in Eq. (3.30). For values of Q2(110) smaller than Q2(101) we

obtained the disclination network shown in Fig. 3.5, however, for Q2(110) > Q2(101) a different,

but similar, texture is found with the four-fold symmetry of the tetragonal axis broken to just

two-fold (not shown).


(a) (b)

Figure 3.5: Textures of the two candidate tetragonal phases; D10+4 (a) and D10−

4 (b). Thedisclinations are shown in blue and the double twist cylinders in grey. The top panel providesa three dimensional view, while the bottom panel shows the projection along the tetragonalaxis. 23 unit cells have been shown for clarity.

Chapter 4

Flexoelectric blue phases

Elastic distortions in the nematic phase of liquid crystals can be categorised into three types:

splay, twist and bend. Stable phases with two-dimensional twist distortions, the cholesteric

blue phases, have been well characterised and observed experimentally. In this Chapter we

describe the occurence of structures with two-dimensional splay-bend distortions.

When nematics are doped with chiral molecules, the chirality of the dopants imparts a

natural twist to the director field. As we have seen in Chapter 2 the texture of these cholesteric

phases is generally a one-dimensional helix, Fig. 4.1(a). However, phases with two-dimensional

helical ordering are also possible, Fig. 4.2(a), and under appropriate conditions some of these

textures can be made stable. These are the blue phases of Chapter 2. One might ask whether

analogous behaviour could be observed in a liquid crystal which shows splay and bend, rather

than twist, distortions.

In 1969 Meyer showed that the one-dimensional splay-bend distortion of the director field,

shown in Fig. 4.1(b), could result from flexoelectric coupling to an external electric field [87].

Here we extend these results to show that near the isotropic-nematic transition two-dimensional

(a) (b)

Figure 4.1: Schematic representation of elastic distortions of nematic liquid crystals, showingthe director field projected into the plane of the page. (a) Twist in the cholesteric phase ofchiral nematics. (b) Splay and bend in a nematic with flexoelectric interactions.

54

4.1 Flexoelectricity 55

(a) (b)

Figure 4.2: Schematic examples of two-dimensional director field configurations. (a) Structuredisplaying regions of local double twist, characteristic of cholesteric blue phases. (b) Analogousstructure possessing double splay distortions.

splay-bend structures, e.g., Fig. 4.2(b), can be stable. We shall term such phases flexoelectric

blue phases.

4.1 Flexoelectricity

The coupling of a liquid crystal to an external field occurs in two principal forms. The most

usual is dielectric, where the field couples directly to the orientational order parameter. This

coupling is quadratic in the field strength and the molecules tend to align parallel or perpen-

dicular to the field, depending on the sign of the dielectric anisotropy. Flexoelectricity, which

we shall deal with here, occurs because of the elastic properties of liquid crystals. If the liquid

crystal is composed of pear- or banana-shaped molecules an elastic distortion can lead to a

spontaneous polarisation. Conversely, if a polarisation is induced by an external electric field

then an elastic distortion results. The polarisation of the liquid crystal reflects the elastic dis-

tortion which accompanies it, taking the form P = esn(∇ · n) − eb(n · ∇)n, and contributing

a term E · P to the free energy density. This is termed flexoelectric coupling, and it gives a

response linear in the field strength [1, 87].

Here, we consider a bulk sample of liquid crystal with periodic boundary conditions so that

all surface terms may be neglected, further assume that the liquid crystal has zero dielectric

anisotropy, so that the effects of the field are purely flexoelectric, and that the electric field

inside the liquid crystal is uniform and equal to the applied external field. Flexoelectricity

involves the coupling of an electric field to gradients in the order parameter, which we briefly

4.2 Landau-de Gennes theory 56

review to define our notational conventions. Up to second order in Q we can construct four

invariants of this form:

Eα∇βQαβ , (4.1)

EαQβγ∇αQβγ , (4.2)

EαQαβ∇γQβγ , (4.3)

EαQβγ∇βQαγ . (4.4)

The first and second of these are total divergences and hence can be discarded, while any

arbitrary linear combination of the last two invariants may be written as

A Eα(Qαγ∇γQβγ +Qβγ∇βQαγ) + B Eα(Qαβ∇γQβγ −Qβγ∇βQαγ) . (4.5)

The first of these terms is a total divergence, Eα∇γ(QαβQβγ), and therefore represents only a

surface electric field coupling. Hence we find that bulk flexoelectricity is given by the unique

coupling

Fflexo = 1V

∫

Ωd3r Qαβ

[

εf(Eα∇γ − Eγ∇α)δβδ]

Qγδ , (4.6)

where V is the volume of the domain, Ω. Analogous arguments have been made some time

ago for flexoelectric coupling expressed in terms of the director field [88]. This is not to say

that the surface terms neglected above are always unimportant. Indeed quite the converse

is true in many cases, particularly when the surface displays topological patterning [89–93].

Nonetheless, here we confine our attention to purely bulk effects and henceforth neglect all

such surface terms.

4.2 Landau-de Gennes theory

The Landau-de Gennes free energy in the presence of both chiral and flexoelectric couplings is

f = 1V

∫

Ωd3r

τ4 tr

(

Q2)

−√

6 tr(

Q3)

+(

tr(

Q2)

)2

+ 27L1

4A0γ

[

(

∇Q)2

+ 4q0Q · ∇ × Q + 2εfL1Qαβ

(

Eα∇γ − Eγ∇α

)

Qβγ

]

,

(4.7)

4.2 Landau-de Gennes theory 57

where we have adopted a dimensionless form for the free energy in a similar manner to that

for cholesteric liquid crystals in Chapter 2. For simplicity, we adopt the one elastic constant

approximation, where the magnitude of twist, splay and bend are taken to be equal. If we take

the view that flexoelectricity acts to induce structure in the liquid crystal in an analogous way

to chirality, then it is natural to again seek stable states by introducing a Fourier decompositon

for Q, Eq. (2.41). When this is done the quadratic part of the free energy takes the form

f (2) =∑

k

N−1k

∑

m1,m2

Qm1(k)Qm2

(k) ei(ψm2(k)−ψm1

(k))M †m1

(k)αβ

τ4δαγ +

27L1

4A0γ

[

k2δαγ − i 4q0 k ǫαδγ kδ − i 2εfEL1

k(

Eαkγ − kαEγ)

]

Mm2(k)βγ .

(4.8)

The basis tensors Mm(k) should be chosen so as to diagonalise f (2) as was done for purely

chiral coupling [60]. It is convenient to introduce the quantity

∆k :=[

1 −(

E.k)2

]1/2, (4.9)

and use the direction of the electric field and of the wavevector for a given Fourier mode to de-

fine a local, right-handed, orthonormal frame field k,v,w such that E =(

E.k)

k + ∆kw. In

terms of this local frame the flexoelectric coupling then takes the form +2ik(εfE∆k/L1) ǫαδγ vδ

revealing a formal mathematical similarity between flexoelectric and chiral couplings. Combin-

ing the two terms motivates a transformation to a new set of basis vectors, obtained by means

of a rotation about w:

a :=1

µk

(

2q0k− εfE∆k

L1v)

,

b :=1

µk

(

εfE∆k

L1k + 2q0v

)

,

c := w ,

(4.10)

where

µk =[

(

2q0)2

+(εfE∆k

L1

)2]1/2

, (4.11)

4.3 Flexoelectric blue phases 58

is a normalisation factor that also serves to define a natural length scale, generalising 2q0 in

the purely chiral case. In this rotated local frame, the choice of orthonormal basis tensors

M±2 = 12

(

b± ic)

⊗(

b ± ic)

,

M±1 = 12

[

a ⊗(

b± ic)

+(

b ± ic)

⊗ a]

,

M0 = 1√6

(

3a ⊗ a − I)

,

(4.12)

results in the desired diagonalisation of f (2),

f (2) =∑

k

N−1k

2∑

m=−2

Q2m(k)

τ4 + 27L1

4A0γ

[

k2 −mµkk]

. (4.13)

We see from Eqs. (4.10) that there is strong sense in which the transition from purely chiral

to purely flexoelectric coupling may be viewed geometrically as a π/2 rotation. Indeed, this is

already evident in the one-dimensional cholesteric and splay-bend states of Fig. 4.1, which are

related by a π/2 rotation about the vertical direction. This rotation is none other than the

well known flexoelectro-optic effect [94].

4.3 Flexoelectric blue phases

Given this formal similarity it is natural to ask whether a precise correspondence can be made

between the stable phases, and their properties, in the purely chiral and purely flexoelectric

limits. To this end we now focus our attention on the purely flexoelectric sector and analyse a

number of structures motivated by the analogy with cholesterics.

The simplest flexoelectric structure is a one-dimensional phase that was described in Meyer’s

original Letter [87], which we shall call splay–bend; see Fig. 4.1(b). The Q-tensor appropriate

to splay–bend is the analogue of that for the cholesteric helix in chiral liquid crystals:

Q =Q2√

2

cos(kx) 0 sin(kx)

0 0 0

sin(kx) 0 − cos(kx)

− Qh√6

−1 0 0

0 2 0

0 0 −1

, (4.14)

where we have used the electric field to define the z-axis of a Cartesian coordinate system and

taken the wavevector to be orthogonal to the field so as to maximise ∆k and hence minimise


the free energy. The free energy is readily shown to be

f = τ4

(

Q22 +Q2

h

)

− 3Q22Qh +Q3

h +(

Q22 +Q2

h

)2+ 27L1

4A0γQ2

2

(

k2 − 2εfEL1k)

. (4.15)

It is formally identical to that of the cholesteric phase with εfE/L1 playing the role of the chiral

parameter, 2q0. Therefore the solutions forQ2 andQh as a function of the reduced temperature,

τ , and field strength, E, obtained by minimising Eq. (4.15), are the same as those obtained

for the cholesteric phase [42,60,82] (Eqs. (2.32) and (2.33)), provided one substitutes 2q0 with

εfE/L1.

Extending the analogy with cholesterics, we now consider flexoelectric phases with a higher

dimensional structure; flexoelectric analogues of the cholesteric blue phases. Based on the

heuristic observations that the field picks out a preferred direction in space, and that the

flexoelectric coupling vanishes for wavevectors parallel to the field, it seems likely that the

free energy will be minimised by two-dimensional structures, containing only wavevectors or-

thogonal to the field. Therefore we consider first a two-dimensional phase with hexagonal

symmetry, obtained by choosing the fundamental set of Fourier modes to be along the direc-

tions ±ex,±(ex+√

3ey)/2,±(ex−√

3ey)/2. An approximate Q-tensor, comprising only this

fundamental set, all with m = 2, and a homogeneous component, is given by

Q =Q2√

6

12

(

ex + iez

)

⊗S e−i(kx−ψ1) + 12

(

−12 ex +

√3

2 ey + iez

)

⊗S e−i(k(−x+√

3y)/2−ψ2)

+ 12

(

−12 ex −

√3

2 ey + iez

)

⊗S e−i(k(−x−√

3y)/2−ψ3) + H.c.

+Qhe

iδ

√6

(

3ez ⊗ ez − I)

,

(4.16)

where H.c. stands for Hermitian conjugate, ψ1, ψ2, ψ3 ∈ [0, 2π), δ ∈0, π and the short-hand

notation ⊗S denotes a symmetrised tensor product, i.e., u⊗S :=u⊗u. The director field of this

hexagonal flexoelectric blue phase is shown in Fig. 4.3. The structure consists of an hexagonal

lattice of strength −1/2 disclinations separating regions in which the distortion is one of pure

splay along any straight line passing through the centre of the hexagon. Indeed, expanding

in cylindrical polars (ρ, φ, z), the director field in a local neighbourhood of the centre of each

hexagon is found to be

n = cos( Q2kρ

3Q2 + 2Qh

)

ez − sin( Q2kρ

3Q2 + 2Qh

)

eρ , (4.17)


Figure 4.3: Director field of the hexagonal flexoelectric blue phase described in the main textobtained from a numerical minimisation at paramter values κE =1, τ−κ2

E =0 where the phaseis stable, see Fig. 4.4. The structure is periodic in both directions.

and therefore the structure may be aptly referred to as a double splay cylinder since it is the

clear analogue of double twist cylinders in the cholesteric blue phases.

The free energy of this hexagonal phase

f = τ4

(

Q22 +Q2

h

)

− κ2

E

4 Q22 − 3eiδ

2 Q22Qh − eiδQ3

h + 2732 cos(ψ1 + ψ2 + ψ3)Q

32

+ 233192Q

42 + 7

2Q22Q

2h +Q4

h − 98 cos(ψ1 + ψ2 + ψ3 + δ)Q3

2Qh ,

(4.18)

where we have defined a flexoelectric analogue of the chirality

κ2E :=

27ε2fE2

A0γL1, (4.19)

is minimised by choosing the phases to satisfy ψ1+ψ2+ψ3 =π and δ=0. As for the splay-bend

phase, this free energy of the flexoelectric phase is identical to the free energy of its cholesteric

analogue, in this case the planar hexagonal blue phase [42, 60]. Consequently, we may draw

on this previous work for cholesterics to conclude that the hexagonal flexoelectric phase has

lower free energy than the splay-bend phase at sufficiently large field strength. However, in

the chiral case the hexagonal blue phase is not stable since one of the cubic blue phases always

has lower free energy. It is therefore necessary to see whether a similar result also holds in the

flexoelectric case.

Although one may expect that the structure of the cubic blue phases provides a good


starting point for a consideration of flexoelectric blue phases, the latter differ in a number

of respects. In particular, since the flexoelectric coupling is vectorial, it defines a preferred

direction which will act to lower the symmetry from cubic to tetragonal. We therefore consider

the set of possible structures obtained by selecting the Fourier modes of Q to be

0,± 2πLx

ex,± 2πLx

ey,± 2πLx

(

ex + ey

)

,± 2πLx

(

−ex + ey

)

,± 2πLz

ez,

± 2πLx

(

ex + Lx

Lzez

)

,± 2πLx

(

ey + Lx

Lzez

)

,± 2πLx

(

−ex + Lx

Lzez

)

,± 2πLx

(

−ey + Lx

Lzez

)

,

(4.20)

where Lx, Lz are the lattice parameters along the x− and z−axes of the conventional unit cell

respectively. This set of structures allows for considerable scope, containing as special cases

Q-tensors that are exact analogues of those representing the cholesteric cubic blue phases as

well as that of the two-dimensional square structure shown in Fig. 4.2(b). We have investigated

the free energy minima of these structures both using approximate analytic calculations and

with an exact numerical minimisation for a range of initial conditions, varying the ratio of the

lattice parameters Lx/Lz and the relative amplitudes and phases of the Fourier components.

The numerical minimisation was performed using a lattice Boltzmann algorithm [32, 34] with

an identical technique to that described in Chapter 2. In all instances a single minimum was

obtained that corresponded to the two-dimensional square flexoelectric blue phase of Fig. 4.2(b),

a result that further supports our heuristic remarks that the suppression of wavevectors parallel

to the field might favour two-dimensional textures.

These results were combined with the analytic expressions for the free energy of the splay-

bend and hexagonal structures, Eqs. (4.15) and (4.18), to construct an approximate phase

diagram. We find that although the two-dimensional square phase is a local minimum of the free

energy within the set of structures defined by Eq. (4.20), it never has lower free energy than the

hexagonal phase. The same result is obtained numerically, leading to the phase diagram shown

in Fig. 4.4, which has been plotted in the κE, (τ−κ2E) plane, in accordance with the conventions

adopted in cholesterics [82], see Chapter 2. As the electric field strength is increased, the

hexagonal flexoelectric blue phase becomes stabilised over an increasing temperature interval

between the isotropic and splay-bend phases. The discrepancy between the numerical and

analytic phase boundaries is similar to that found in cholesterics and arises from neglecting

higher order wavevector harmonics in analytic calculations. In the cholesteric blue phases the

lattice periodicity is not identical to the cholesteric pitch, but is generally somewhat larger.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

−1.5

−1

−0.5

0

0.5

1

κE

τ −

κE2

Isotropic

Splay−−bend

Hexagonal

Figure 4.4: Phase diagram of the flexoelectric blue phases. The circles and solid lines show thenumerical phase boundaries, while the analytic results are given by the dashed lines.

From the numerical minimisation we find the same feature in the hexagonal flexoelectric blue

phase, with the lattice parameter being approximately 15% larger than the ‘pitch’ (πL1/εfE)

of the splay-bend phase at the triple point.

The phase diagram bears a qualitative resemblance to that of cholesterics (Chapter 2),

but the details reveal differences between chiral and flexoelectric couplings. In particular, it

seems that the suppression of wavevectors parallel to the field introduced by the quantity ∆k is

sufficient to convert the fully three-dimensional states which are stable at large chiral coupling

into two-dimensional states under flexoelectric coupling.

We comment that the local director field in the double splay regions of the flexoelectric

blue phases is the same as that in the escape configuration lattice phase recently suggested by

Chakrabarti et al. [8]. However, the mechanism by which the structure occurs, and is stabilised,

is different. Their proposed structure arises without the application of an electric field, being

instead stabilised by the saddle splay elastic constant and including weak anchoring of the

director field and surface tension at the interface between the nematic and isotropic regions of

the fluid. Note also that the hexagonal symmetry of the phase we obtain is different to the

square symmetry predicted in that work.

4.3.1 Experimental estimates

It is important for potential experimental work to estimate the field strength required to sta-

bilise the two-dimensional hexagonal flexoelectric blue phase. A good estimate is that the

4.4 Flexoelectric effects in the cholesteric blue phases 63

strength of the flexoelectric coupling, εfE, should be as large as the chiral coupling, 2q0L1,

in a compound which possesses cholesteric blue phases. This gives E ≈ 2q0L1/εf ≈ πK/ep,

where we have replaced the Landau–de Gennes parameters with their director field equivalents;

an average Frank elastic constant, K, and the average of the flexoelectric coupling constants,

e :=(es + eb)/2. p is the pitch in the cholesteric phase of a compound which also displays blue

phases, typically ∼ 0.3µm. Recently materials have been developed for flexoelectric purposes

and found to have a large flexoelastic ratio, e/K ∼ 1V −1 [95]. Thus in these materials the

required field strength is expected to be E∼10V µm−1. It will also be helpful to use a material

with close to zero dielectric anisotropy and to take care to ensure that surface effects are neg-

ligible compared to the bulk, as we have assumed here. The analogy with cholesterics further

suggests that flexoelectric blue phases are only likely to be found in a narrow temperature range

(∼1 K) just below the isotropic-nematic transition temperature, although this is predicted to

expand with increasing field strength.

4.4 Flexoelectric effects in the cholesteric blue phases

The Landau-de Gennes theory we developed in § 4.2 showed that chiral and flexoelectric cou-

plings could be described within a common framework, the transition from one limiting case

to the other occuring via a π/2 rotation. For one-dimensional textures the cross-over from the

cholesteric helix to the flexoelectric splay-bend is smooth, without any kind of sharp transition.

However, this smooth cross-over does not extend to the blue phases, for on the chiral side the

stable textures are three-dimensional, while on the flexoelectric side they are two-dimensional.

Thus at some intermediate stage a transition must occur. In this final section we illustrate this

transition for two specific cases: first we will look at blue phase I with a field applied parallel

to the [011] direction, and then we will consider blue phase II with a field applied parallel to

[001], thereby mirroring our analysis of dielectric switching in Chapter 3. We begin with a brief

review of the cross-over between one-dimensional textures.

4.4.1 Flexoelectro-optic effect in cholesterics

If a cholesteric liquid crystal is prepared with its helical axis parallel to the cell plates (uniform

lying helix) the cell appears uniformly dark when viewed between crossed polarisers. The cell

can be made to transmit light by applying a small electric field perpendicular to the helical


axis, indicating a rotation of the optical axis, and extinction subsequently regained by a small

rotation of the cell. The rotation angle needed to regain extinction is found to be linear in

the applied field strength, for small fields, and to change sign when the direction of the field is

reversed. This effect, known as the flexoelectro-optic effect, was first described by Patel and

Meyer [94] and has since been developed for potential display technologies [95–97].

If we assume a minimal modification for the Q-tensor, then the cholesteric under an applied

field will be described by

Q = Q2√2

[

(

b⊗ b − ez ⊗ ez)

cos(kx) +(

b⊗ ez + ez ⊗ b)

sin(kx)]

− Qh√6

[

3a ⊗ a − I]

, (4.21)

where we have taken the helical axis to define the x-axis and then the direction of the elec-

tric field to define the xz-plane. This Q-tensor provides a homotopy between the zero field,

cholesteric (Eq. (2.25)), and the zero chirality, splay-bend (Eq. (4.14)), textures. As such, the

effect of flexoelectric coupling on the cholesteric phase is to rotate an initially chiral texture

into an achiral one. The angle of rotation, as a function of field strength, is given by

tan(φ) = εfE∆k

2q0L1, (4.22)

in agreement with the result first obtained in Reference [94]. This rotation angle provides both

a simple, geometrical view of the distortion brought about by the field and also a natural,

and convenient, dimensionless measure of the relative strength of the chiral and flexoelectric

couplings. Recently rotation angles of as much as 87o have been reported [95].

4.4.2 Blue phase I

When an electric field is applied to a cholesteric blue phase it leads to both a distortion of the

director field of the texture and also to a change in the size and shape of the unit cell. At large

field strengths the distortion and change in shape can be appreciable, ultimately leading to field-

induced textural transitions. For the same reasons as discussed in Chapter 3, the change in size

and shape is quadratic in the field strength at low fields and may be described by a change in

the lattice vectors of the unit cell through a fourth rank electrostriction tensor. However, unlike

the analogous case of dielectric coupling, nothing is currently known about the electrostriction

tensor arising from flexoelectric effects: neither the typical magnitude, nor signs of R1, R2 and


Figure 4.5: Flexoelectric switching of blue phase I for fields parallel to [011]. At small fieldstrengths the unit cell distorts to become tetragonal, resulting in a textural transition to acentred tetragonal phase with the same space group as blue phase X (see § 3.3). At larger fieldstrengths a second transition is found to a simple tetragonal phase with space group P4222. Atstill larger fields there is a final transition to the two-dimensional hexagonal flexoelectric bluephase. The disclination lines are shown in blue and 23 unit cells have been shown for clarity.

R3. Nonetheless, the eventual transition to the two-dimensional hexagonal flexoelectric blue

phase and suppression of wavevectors parallel to the field brought about by the quantity ∆k

suggest that the texture should undergo an expansion parallel to the field direction, in which

case both R1 and R3 would be positive. Preliminary numerical simulations indicate that this

is the case and that when the field is applied parallel to the crystallographic [011] direction, it

promotes a distortion towards a tetragonal geometry, as is also found under dielectric coupling.

Therefore, this is the geometry we employ to look at the chiral-to-flexoelectric cross-over in

blue phase I.

Blue phase I is initialised in a tetragonal unit cell of size 32×32×40 with its [011] direction

parallel to the z-axis of the simulation unit cell. The texture is allowed to relax for a short

period before an electric field is applied parallel to the z-axis. As was found for dielectric

effects in Chapter 3, the electric field induces a twist in the disclination lines allowing them to

transiently merge, thus fascilitating textural transitions. These are illustrated in Fig. 4.5, with

the field strength at which the transitions occur quoted in terms of tan(φ), the dimensionless

measure of the relative strength of chiral and flexoelectric couplings. For the largest applied

field strengths a transition is indeed observed to the two-dimensional hexagonal flexoelectric


Figure 4.6: Flexoelectric switching of blue phase II for fields parallel to [001]. The disclinationlines are shown in blue and the double twist cylinders in grey. In all cases 23 unit cells havebeen shown for clarity. For the hexagonal flexoelectric blue phase we show the director fieldconfiguration viewed along the field direction and show 22 simulation unit cells. The hexagonalsymmetry of the preferred texture is evident despite the imposed square symmetry of thesimulation box.

blue phase, as expected. However, at intermediate values, two further transitions precede it,

each yielding distinct textures that are stable over a small range of field strengths. The first

it to a centred tetragonal texture with space group I4122, which we have called blue phase FI

and that we recognise as having the same disclination network as the blue phase X structure

found under dielectric switching and described in § 3.3.1. At slightly higher field strengths

there is a second transition to a distinct tetragonal texture that we have called blue phase FII,

this time with space group P4222. In this texture, the disclinations are all parallel to the field,

occuring in pairs that wrap around each other to form a double helix, and with the axes of

the double helices themselves then arranged on a square lattice. Finally, the two members of

a given double helix transiently merge and re-separate, allowing them to both straighten out

and to adopt an hexagonal configuration in the plane perpendicular to the field.

4.4.3 Blue phase II

We now follow a similar analysis, but for blue phase II and with the field applied parallel to the

crystallographic [001] direction. The sequence of transitions is shown in Fig. 4.6. At relatively

small field strengths there is an unbinding of the bicontinuous network of disclinations, in a


similar manner to that found under dielectric coupling in § 3.2, but with the unbinding occuring

in a different direction. This is accompanied by a breaking of the double twist cylinders which

lie initially perpendicular to the field. After this unbinding the disclinations form pairs of

double helices arranged on a square lattice, which we identify with the texture blue phase FII

also found in the switching of blue phase I. Finally, at field strengths of tan(φ) ∼ 1.4 there is

a further transition to a two-dimensional texture corresponding to the hexagonal flexoelectric

blue phase.

Double twist cylinders

We conclude this Chapter with a brief analysis of the small field distortion of blue phase II,

focusing on the changes induced in the double twist cylinders. We have seen in § 2.5 that

an approximate Q-tensor containing only the fundamental set of wavevectors is sufficient to

correctly describe the blue phase texture. For small field strengths we can assume that the

principal effects of the electric field are captured by the same approximate Q-tensor, but with

the basis tensors of Eq. (4.12), which are appropriate to flexoelectric coupling,

Q =Q2(k⊥)√

4

(

εfEµL1

)2cos(k⊥x)

2q0εfEµ2L1

cos(k⊥x)εfEµL1

sin(k⊥x)

2q0εfEµ2L1

cos(k⊥x)(2q0µ

)2cos(k⊥x)

2q0µ sin(k⊥x)

εfEµL1

sin(k⊥x)2q0µ sin(k⊥x) − cos(k⊥x)

− Q2(k⊥)√4

(2q0µ

)2cos(k⊥y)

−2q0εfEµ2L1

cos(k⊥y)−2q0µ sin(k⊥y)

−2q0εfEµ2L1

cos(k⊥y)(

εfEµL1

)2cos(k⊥y)

εfEµL1

sin(k⊥y)

−2q0µ sin(k⊥y)

εfEµL1

sin(k⊥y) − cos(k⊥y)

+Q2(k‖)√

2

cos(k‖z) sin(k‖z) 0

sin(k‖z) − cos(k‖z) 0

0 0 0

.

(4.23)

Here, we have also allowed for the electric field to lower the symmetry from cubic to tetragonal

by assigning different wavevector magnitudes and Fourier amplitudes to the modes parallel

and perpendicular to the field. The limit E → 0 is recovered by requiring k⊥ → k‖ and

Q2(k⊥) →√

2Q2(k‖).

In the cholesteric texture, the effect of the electric field could be simply interpreted as a

rotation of the director about the z-axis. We shall show that the distortion of the double twist


cylinders in blue phase II may be similarly described in terms of a rotation of the director. Two

distinct cases need to be considered: cylinders whose axes are initially parallel to the field, and

cylinders whose axes are initially perpendicular to the field. Starting with the former case, we

can expand the Q-tensor about the line (1/2, 0, z) to obtain

Q =Q2(k⊥)√

4

−1 0 − εfEk⊥δxµL1

+ 2q0k⊥δyµ

0 −1 − εfEk⊥δyµL1

− 2q0k⊥δxµ

− εfEk⊥δxµL1

+ 2q0k⊥δyµ − εfEk⊥δy

µL1− 2q0k⊥δx

µ 2

+Q2(k‖)√

2

cos(k‖z) sin(k‖z) 0

sin(k‖z) − cos(k‖z) 0

0 0 0

.

(4.24)

If we similarly expand the director in a neighbourhood of the cylinder axis as n = (nx, ny, 1)

then a short calculation reveals that

nx

ny

= Q2(k⊥)

9Q22(k⊥)−2Q2

2(k‖)

−√

2Q2(k‖) sin(k‖z) 3Q2(k⊥) +√

2Q2(k‖) cos(k‖z)

−3Q2(k⊥) +√

2Q2(k‖) cos(k‖z)√

2Q2(k‖) sin(k‖z)

×

2q0µ

εfEµL1

−εfEµL1

2q0µ

k⊥δx

k⊥δy

.

(4.25)

Thus the distortion of the cylinder can indeed be viewed as a rotation about the z-axis, again

through an angle φ = arctan(εfE/2q0L1), as in the cholesteric flexoelectro-optic effect. How-

ever, whereas for the cholesteric the distorted director is obtained by rotating the local director

without displacing it from its initial position, we see here that the distorted double twist cylin-

der at a point δr := (δx, δy) is given by the undistorted configuration at the rotated point

Rδr, with R a rotation matrix through angle φ. Thus the effect of the field is to induce a bulk

rotation of the entire double twist cylinder.

The situation turns out to be quite different for cylinders whose axes are initially perpen-

dicular to the field. One such axis, described in § 2.5, is the line (x, 1/2, 0), defined by the locus

of points for which the director is n = (1, 0, 0). In non-zero field one may readily verify that

the line (x, 1/2, 0) no longer describes the cylinder axis, which instead is given by solutions of


the simultaneous equations

Q2(k⊥)2

(

εfEµL1

)2cos(k⊥x) − Q2(k⊥)

2

(

2q0µ

)2cos(k⊥y) +

Q2(k‖)√2

cos(k‖z) = S , (4.26)

Q2(k⊥)2

2q0εfEµ2L1

(

cos(k⊥x) + cos(k⊥y))

+Q2(k‖)√

2sin(k‖z) = 0 , (4.27)

εfEµL1

sin(k⊥x) + 2q0µ sin(k⊥y) = 0 , (4.28)

where S is the maximal eigenvalue of Q. The last of these equations may be simply rearranged

to give

sin(k⊥y) = − tan(φ) sin(k⊥x) . (4.29)

This is an important result, which shows that the axis only remains continuous as one moves

along the x-direction if tan(φ) < 1, i.e., so long as the chiral coupling remains larger than the

flexoelectric coupling. For field strengths above this value the cylinders perpendicular to the

field must necessarily break. Of course, this estimate provides only an upper bound and it is

perfectly plausible that the cylinders will break at much lower field strengths.

For small field strengths, tan(φ) ≪ 1, we can expand about the zero field solution, letting

the cylinder axis be given by (x, 1/2 + δy, δz), whereupon we find

δy = 12π tan(φ) sin(k⊥x) , (4.30)

δz = 1π tan(φ) sin2(k⊥x/2) . (4.31)

Note that while the displacement perpendicular to the field, δy, averages to zero over the unit

cell, the displacement parallel to the field, δz, does not. We might say that the mean position

of the cylinder has been raised in the direction parallel to the field by an amount linearly

proportional to the field strength, at least for small fields. This mean position is given by the

line (x, 1/2, (1/2π) tan(φ)), which the cylinder axis wraps around in a helical fashion.

The distortion and breaking of double twist cylinders that we have just described is in good

qualitative agreement with the results of a full numerical simulation shown in Fig. 4.6.

Chapter 5

Locomotion: low Reynolds number swimmers

The ability to move is a ubiquitous trait. From birds in the sky to wildebeast on the plains of

the Serengeti and cyclists on the streets of Oxford, living things are able to generate their own

motion. Understanding this motion has long been a topic of interest to biologists, mechani-

cal engineers and applied mathematicians, but recently it has also attracted growing interest

from condensed matter physicists, primarily because of the remarkable properties of collective

locomotion in flocks and herds [98].

Systems as seemingly unrelated as flocks of birds, swarms of bacteria and layers of vibrated

granular rods all display features of long, or quasi-long, range order and large scale hydrody-

namic swirls and jets, leading to their continuum description within the common theoretical

framework of active ordered fluids [98,99]. The challenge and interest to the condensed matter

physicist is, as always, to ask: how can this long range cooperativity and collective behaviour

be understood in terms of the interactions between individuals?

At the modelling level, many of the common features can be captured by the simple rules;

follow and avoid [100]. Each individual moves in the average direction of the others it can ‘see’,

while at the same time trying to avoid collisions with any of its neighbours. However, while

birds, or people, might enact these rules (semi-)literally, it is more difficult to imagine that the

same is true of much simpler organisms like bacteria, for which similar collective behaviour is

observed [101–103]. Although it is known that bacteria can detect and respond to chemical

gradients [104–106] and to obstacles in the fluid [107] it seems clear that the level at which they

do this is different to that of a bird, or wildebeast. Such a simple organism should derive its

response from simple principles. In this regard, recent research has focused on the role played

by hydrodynamic interactions in determining the collective motion of bacteria and other simple

70

71

Figure 5.1: Examples of microscale locomotion. (a) Echerichia coli bacteria swim by rotatinghelical flagella, from Reference [128]. (b) Water striders move at the surface of a liquid, ex-ploiting surface tension for their motility, from Reference [124]. (c) Collective swarming in asessile drop of Bacillus subtilis, from Reference [103].

self-propelled particles [99,108–113].

The means by which living things achieve their locomotion vary greatly depending on their

environment (Fig. 5.1). While fish are able to swim by waving their tails much like a rudder,

such motions are not successful for very small organisms such as bacteria, or sperm [114,115].

This is because the inertial forces which provide the thrust for a fish are vastly overwhelmed

by the effects of viscous stresses in the case of bacteria. When an organism of characteristic

length L moves at speed U through a fluid of viscosity µ and density , the ratio of the inertial

and viscous forces defines the Reynolds number for the flow

Re =UL

µ. (5.1)

For most fish the Reynolds number is of the order of several thousands and inertia is important,

while for micron sized bacteria it is more typically ∼ 10−5 and inertia is unimportant. For such

small Reynolds numbers the inertial terms in the Navier-Stokes equations may be neglected

and the fluid motion adequately described by the Stokes equations

µ∇2u −∇p = 0 , (5.2)

∇ · u = 0 , (5.3)

where p is the pressure in the fluid. With the explicit time dependence of the inertial terms

gone, the flow is specified throughout the fluid by the instantaneous motion at the boundaries.

The distinctive properties of viscous fluid motion were beautifully illustrated by Sir Geoffrey

72

Taylor in his educational film “Low Reynolds number flows” [116]. In one experiment a small

amount of dye was added to a viscous fluid contained between two coaxial cylinders. When the

outer cylinder was rotated by four turns in the clockwise sense, the dye was smeared through

the fluid. However, upon counter-rotating the cylinder by four turns in the anticlockwise sense,

the blob of dye was exactly recovered as it had been initially. This, now famous, experiment

demonstrates the reversible nature of low Reynolds number flows. Taylor went on to vividly

demonstrate the consequences of this reversibility for the propulsion of microscopic organisms

by showing that a simple model swimmer utilising a waving rudder is unable to swim in a

viscous liquid.

The fact that reciprocal motions do not lead to swimming at low Reynolds number was

popularised by Purcell [117] and is commonly referred to as the Scallop theorem. Purcell

emphasised further that the speed with which the motion is performed is unimportant. Waving

the rudder rapidly in one direction and very slowly in the other does not alter the result: at

the end of a complete stroke the model returns exactly to its initial position. This is also true

for things that actually swim. The distance that a swimmer moves in one complete swimming

stroke is unchanged if the swimmer performs the same stroke twice as fast. Only the sequence

of shapes matters [118–120].

Since Taylor’s seminal work [114,115] there has been steady progress in our understanding

of the propulsion of individual swimmers [119, 121–125] and in developing optimal swimming

strategies [120,126]. The importance of rotational motions was cemented by Berg and Ander-

son who demonstrated that many bacteria, such as Escherichia coli, swim by the rotation of

flagella [127, 128]. The propulsive hydrodynamics of rotating helical filaments was described

in detail by Lighthill in his 1975 John von Neumann Lecture [121] and later extended by

Higdon [129, 130]. Within the past decade the theory of the swimming of elastic filaments

has been developed [123, 131] and applied to describe periodic chirality reversals in bacterial

flagella [132,133] and the entire cell bodies of Spiroplasma [134,135].

Considerable insight in recent years has been gained by the analysis of simple models that

capture the essence of swimming whilst remaining analytically tractable, following the examples

laid down by Taylor [115] and Purcell [117]. This has included both a more detailed analysis of

the models proposed by Taylor and Purcell [136–139] and the proposal of new models based on

a small number of linked spheres [140–145] or phoretic effects generated by chemical reactions

at the swimmer’s surface [146,147]. Initial experiments are beginning to show the viability of

73

some of these simple models as the basis for artificial microswimmers that may be developed

for drug delivery or manipulating payloads in microchannels [148–150].

However, whilst our understanding of the motility of single organisms has developed signif-

icantly, much less is known about the detailed hydrodynamics of swimmers; the time averaged

flow fields that they generate and the interactions that occur between individuals. These inter-

actions may be expected to be substantial because of the long range 1/r nature of the fluid flow

generated by point forces at low Reynolds number. They have been shown to be important

in the gyrotactic focusing of bottom heavy algal cells [151], band formation in magnetotactic

bacteria [152,153] and in many aspects of bacterial behaviour near surfaces [154–157].

The flow field generated by a swimmer is often described as being dipolar at large dis-

tances [101, 109, 110]. This is because swimmers are self-motile and are not dragged through

the fluid by external influences, so that, at every instant, they must experience zero resultant

force [115, 121]. A multipole description of the flow produced by a swimmer will thus have as

its most slowly decaying part a dipolar term, varying with distance as 1/r2. The dipolar nature

of the far field flow generated by a swimmer seems to be so widely accepted that it is rarely

calculated explicitly. A noteable exception is a detailed discussion by Lighthill of the flow field

generated by a rotating helical filament [158] in which he demonstrated that the combination

of finite length and zero resultant torque on the helix was sufficient to imply that the far field

flow decays more rapidly than 1/r2. Moreover, in the case of Spirochetes, Lighthill showed that

“its three-dimensional flow field remains spatially concentrated - without any alge-

braically decaying terms in the far field - even when the helix is idealised as one of

unbounded extent” [158].

Even in the case when the far field flow does possess a dipolar contribution, it is unclear

exactly how far from the swimmer one needs to be before this dipolar character is evidenced.

Nor is there presently any information on how this distance depends on the type of swimmer,

or the detailed properties of its swimming stroke. A naive expectation might be that the

magnitude of the far field flow should scale with the slenderness of the swimmer and the

amplitude of the swimming stroke in the same way as the swimming speed, which is in general

linearly and quadratically, respectively [114, 119], and that this should apply to all terms in a

multipole expansion. The transition between near and far field fluid flow would then occur at

a distance determined by purely numerical factors, which might even be generic across many

74

different types of swimmer. We shall show in Chapter 6 that these naive expectations are not

borne out for a particular model of linked-sphere swimmers.

Hydrodynamic interactions between swimmers result in each individual being both advected

and rotated by the fluid flow generated by the others. It might be thought, therefore, that

the interactions will assume the same dipolar far field form as is expected of the fluid flow,

an approximation that is often taken as the basis for the development of continuum models of

swimmers [99, 108, 109]. However, given that the far field form of the flow is not necessarily

dipolar we should perhaps be more cautious in assuming any particular behaviour for the

hydrodynamic interactions.

Another common approach is to treat the swimmer only in an effective sense, imagined in

its simplest form as a ‘body’ and a ‘thruster’ [110–113]. In doing so, these approaches do not

fully resolve the details of the swimmer and make no attempt to describe its swimming stroke.

Thus there is no sense in which a distinction between transverse and longitudinal modes of

propulsion can be made. The shape of the swimmer is also neglected and yet our experience

with liquid crystals leads us to expect that long, thin, rod-like swimmers, like that introduced

by Purcell [117], will have different collective behaviour than flat, disc-like swimmers, such as

the one described by Taylor [115].

More faithful descriptions of the hydrodynamic interactions between two swimmers have

been studied using a variety of theoretical models [159–168]. Some of the earliest studies

made use of flagella driven micromachines [159, 160] and looked at the propulsive advantage,

or disadvantage, of side-by-side and tandem swimming. The interactions between two fixed,

parallel, rotating helices were investigated [161], revealing that they do not phase synchronise

when driven at constant torque. Ishikawa and co-workers [162–164] have studied in detail a

simple ‘squirmer’ model, matching far field asymptotics onto near field lubrication theory and

recently applied their results to the description of collective behaviour in small groups of up to

two hundred individuals [165].

Nonetheless there remains a sizeable gap in our understanding of how the specific properties

of a given swimmer and its swimming stroke influence the hydrodynamic interactions between

them. Even many simple questions remain largely unanswered: are the interactions always

attractive, or does this depend on the relative positions of the two swimmers? Does the

relative phase of the two swimmers play an important role? How important are the details of

the particular swimming stroke in determining the interactions? Or, equivalently, how generic

5.1 T-duality 75

are the hydrodynamic interactions between swimmers?

In the remainder of this thesis we shall develop a description of the hydrodynamics of a

simple model of linked-sphere swimmers first introduced by Najafi and Golestanian [140] and

attempt to answer some of these questions. In Chapter 6 we shall formulate our hydrodynamic

description and analyse in some detail both the time averaged flow field generated by a single

swimmer and the form, and origin, of the hydrodynamic interactions between individuals.

We apply our results to two specific cases; the long time trajectories of two initially parallel

swimmers at the end of Chapter 6 and swimmer scattering in Chapter 7.

Throughout our analysis the presence will be felt of the most overarching facet of low

Reynolds number hydrodynamics: the kinematic reversibility of Stokes flows. The importance

of this has long been recognised for the motility of an individual [114–117], but its consequences

for interactions between swimmers and collective motility are only beginning to come to light.

We conclude this Chapter with a description of time reversal symmetry and swimming, leading

to an important general result concerning the flow field produced by a swimmer, which we shall

call Pooley’s corollary.

5.1 T-duality

An important concept in understanding the swimming of microscopic organisms is that the

Stokes equations, which govern zero Reynolds number fluid flows, do not possess any intrinsic

notion of time. This leads directly to the concept of T-duality, which we now describe. Consider

any successful swimming stroke A: the time reversed swimming stroke A corresponds to an

equally successful stroke of the same organism. We refer to this as the T-dual swimmer, which

may be interpreted as the original swimmer going backwards in time.

As an example, consider a simple model of E. coli in which a spherical cell body is propelled

by a helical flagellar bundle. The bacterium has a rotary motor that causes the flagella to

rotate [104, 127, 128], thereby generating propulsion with the cell body being pushed through

the fluid by the rotation of the flagellar bundle lying behind it. We might say that E. coli is an

example of a “pusher”. If we view this motion backwards in time, then we would observe that

swimming is still generated by the rotation of the flagella, but that instead of pushing the cell

body now the flagellar bundle is pulling it from in front. Thus we would call this swimmer a

“puller”. T-duality is merely the observation that pullers are pushers going backwards in time

5.1 T-duality 76

Figure 5.2: Schematic illustration of T-duality. (a) A pusher, such as E. coli, viewed backwardsin time is a puller. (b) Some swimmers, such as Spirillum volutans, look the same forwardsand backwards in time.

(Fig. 5.2).

Exactly the same comments can be made about extensile and contractile swimmers. This

terminology derives from the nature of the time averaged flow field produced by the swimmer.

At large distances the average flow will, in many cases, be well approximated by a dipolar field,

decaying with distance as 1/r2. If the fluid flow is away from the swimmer along its direction of

motion then the swimmer is said to be extensile, while if the fluid flow is towards the swimmer

then it is called contractile. T-duality provides a relationship between the two: a contractile

swimmer may be interpreted as an extensile swimmer going backwards in time.

A special group of swimmers are distinguished by having swimming strokes that are un-

changed when viewed backwards in time. We refer to such swimmers as self T-dual. It is

worth being precise here to avoid any potential confusion with reciprocal swimming strokes. A

reciprocal stroke is one that is time reversal invariant. Such motions do not lead to swimming

on account of kinematic reversibility [116,117,119,169]. A self T-dual stroke on the other hand,

is one that is time reversal covariant. Several commonly used simple model swimmers are self

T-dual, for example a sinusoidally waving sheet [114], Purcell’s three link swimmer [117], the

Najafi-Golestanian swimmer [140], the pushmepullyou swimmer [144] and the twirler (rotating

doughnut) [115,138,139]. Moreover, the rigid rotation of a helical filament [121] is an example

of a self T-dual swimming stroke, so that a number of bacteria such as Spirillum volutans and

Spirochetes belong to this class to a very good approximation.

5.1 T-duality 77

5.1.1 Pooley’s corollary

In the case of self T-dual swimmers we can deduce an immediate corollary about the nature of

the fluid flow field that they generate. Since their swimming stroke is unchanged when viewed

backwards in time, their flow field must be too. Therefore, at large distances, it can be neither

extensile, nor contractile, since these look different under time reversal. Thus we conclude that

the far field flow of a self T-dual swimmer is not dipolar, but rather decays more rapidly and

at least as fast as 1/r3.

We shall see an explicit example of this result when we analyse the hydrodynamics of the

Najafi-Golestanian swimmer in Chapter 6, however, in agreement with these general symmetry

arguments, Lighthill has already shown that the far field flow produced by a helical filament

decays more rapidly than 1/r2 [158].

Chapter 6

Hydrodynamics of linked sphere model swimmers

6.1 The Najafi-Golestanian swimmer

We now describe a simple model low Reynolds number swimmer that was first introduced by

Najafi and Golestanian [140] and which we shall take as the basis for much of our analysis. The

swimmer consists of three spheres, each of radius a, aligned colinearly and connected by thin

rods of ‘natural’ length D. The rods are made to extend and contract, e.g., by the imagined

action of an internal motor, in a periodic and non-time reversible manner. In the original

construction, shown in Fig. 6.1, the swimming stroke was broken up into four separate stages,

during which the length of only one of the rods was altered. The spheres are labelled 1, 2 and

3, as indicated in Fig. 6.1, increasing in the direction n, parallel to its long axis and along the

direction in which it swims. The hydrodynamics of the model is greatly simplified if we assume

that the three spheres are small and far apart, a/D ≪ 1, and that the rods connecting them

are sufficiently thin for their hydrodynamic effect to be neglected. Such sphere plus ‘phantom’

linker models have become increasingly popular in recent years [144,145,166].

In this Chapter we will study a variant of this model in which the four stage swimming

stroke is replaced by sinusoidal oscillations of the arm amplitudes. Specifically, the length of

the rear and front arms are taken to be

D + ξR sin(

ωt)

, and D + ξF sin(

ωt− φ)

, (6.1)

respectively. Here, we are allowing for the amplitudes of the two oscillations to be different, a

feature of some importance. Reversing the direction of time, one sees that the T-dual swimmer

78

6.2 Hydrodynamics of a single swimmer 79

Figure 6.1: Illustration of the Najafi-Golestanian swimmer. The left box shows the originalfour stage swimming stroke described in Reference [140]. The right box shows the sinusoidalversion that we consider here.

corresponds to an interchange of the amplitudes, ξR ↔ ξF, and that the swimmer is self T-dual

if the amplitudes are equal. Breaking this symmetry by allowing the two arm amplitudes to

be different leads to qualitative changes in the hydrodynamic interactions between swimmers.

Throughout this Chapter we employ a short-hand notation

ξR := ξR sin(

ωt)

, (6.2)

ξF := ξF sin(

ωt− φ)

, (6.3)

to reduce the length of already lengthy formulae.

6.2 Hydrodynamics of a single swimmer

The linearity of the Stokes equations allows the velocity field generated by any low Reynolds

number flow to be written as a linear combination of the forces acting on the fluid

u(x) =∑

r

Gr(x)f r , (6.4)

where f r is the force acting on sphere r of the swimmer and Gr(x) is the appropriate Green

function for the fluid domain under consideration. The problem in applying Eq. (6.4) in practice

is that neither the forces, nor the Green function are known exactly. A common approximation,

relevant to a dilute suspension of spherical objects in an unbounded fluid, is to take the Green


function to be the Oseen tensor [170]

Gr(x) =

16πµa I if |x− xr| = a ,

18πµ

1|x−xr|

(

I + (x−xr)⊗(x−x

r)|x−xr|2

)

otherwise ,

(6.5)

where µ is the fluid viscosity and a is the sphere radius. We shall adopt this approximation for

the remainder of this thesis.

The fluid motion is therefore entirely determined by a knowledge of the forces f r with which

the swimmer acts on the fluid as a result of its changing shape. These are determined by the

combined requirements that each swimmer generates no net force,∑

r fr = 0, and that the

flow field produced is consistent with the shape changes that it undergoes during its swimming

stroke

u(x2) − u(x1) =: bR =(

∂tξR)

n , (6.6)

u(x3) − u(x2) =: bF =(

∂tξF)

n . (6.7)

Inserting Eq. (6.4) for the fluid flow converts these consistency requirements into a relationship

between the changing shape of the swimmer and the forces with which it acts on the fluid

16πµaI

(

−2 −11 2

)

+ G1(x2)(

2 1−1 0

)

+ G2(x3)(

0 1−1 −2

)

+ G1(x3)(

0 −11 0

)

(

f1

f3

)

=(

bR

bF

)

, (6.8)

where we have eliminated f2 using the force-free constraint. In our approach the shape of the

swimmer is specified as a function of time and the forces are the unknowns, which are deter-

mined by inverting Eq. (6.8). The simple linear geometry of the Najafi-Golestanian swimmer

allows this inverse to be calculated exactly, however, in anticipation of our subsequent calcula-

tions we will not pursue this, and instead give a perturbative inversion, using a/D as a small

parameter

(

f1

f3

)

= 2πµa

bR(

−21

)

+ bF(

−12

)

− 2πµa[

G1(x2)

bR(

41

)

+ bF(

−12

)

+ G2(x3)

bR(

−21

)

+ bF(

−1−4

)

+ G1(x3)

bR(

4−5

)

+ bF(

5−4

)]

+ o(

[a/D]2)

. (6.9)


6.2.1 Translational motion

We now describe the translational motion of the swimmer through the fluid by determining

its position as a function of time. Once the position of any one of the spheres is known, the

positions of the others follow from the prescibed shape of the swimmer, hence it suffices to

keep track of only one of them. We choose to track the position of the centre sphere, although

this choice is somewhat arbitrary and any other choice would be just as good.

In the zero Reynolds number limit the velocity of each sphere is required to match instanta-

neously the local velocity of the fluid surrounding it, so that the position of the sphere evolves

according to

dx2

dt= u(x2) , (6.10)

= 13

(

bR − bF)

+ 2πµa3

G2(x3)[

2bR + bF]

− G1(x2)[

bR + 2bF]

+ G1(x3)[

bF − bR]

+ o(

[a/D]2)

.

(6.11)

Integrating this expression gives the displacement of the centre sphere from its initial position

as

δx2(t) = n

∫ t

0dt′

13 ∂t′

(

ξR − ξF)

+ a6

[

1D+ξF

∂t′(

2ξR + ξF)

− 1D+ξR

∂t′(

ξR + 2ξF)

+ 12D+ξR+ξF

∂t′(

ξF − ξR)

]

+ o(

[a/D]2)

.

(6.12)

The first term describes an o(ξ) oscillation about its initial position that integrates to zero

over a complete swimming stroke. This is precisely the motion the sphere would undergo if the

swimmer was placed in vacuum and corresponds to the centre of mass of the swimmer remaining

fixed. Since the swimmer is in a fluid, and not in vacuum, this is not the only contribution to its

motion. The additional terms at o(a/D), and higher, arise due to hydrodynamic interactions

and describe swimming, since they do not integrate to zero over a complete swimming stroke.

Performing the integration we find that to leading order the total distance moved after each

stroke is

T = 2πa3

ξRξF sin(φ)D2

(D/ξR)2[

(

1 − (ξR/D)2)−1/2 − 1

]

+ (D/ξF)2[

(

1 − (ξF/D)2)−1/2 − 1

]

− 14(2D/Ξ)2

[

(

1 − (Ξ/2D)2)−1/2 − 1

]

+ o(

[a/D]2)

,

(6.13)


where

Ξ =(

(ξR)2 + 2ξRξF cos(φ) + (ξF)2)1/2

. (6.14)

6.2.2 Time averaged flow field

The motion of the swimmer constitutes only one part of its hydrodynamics. At low Reynolds

numbers fluid disturbances often extend appreciable distances from their point of origin, ex-

ercising considerable influence over other objects in the flow. The net flow field generated by

a swimmer over the course of its swimming stroke is one such disturbance, whose influence at

large distances we now describe.

The time averaged flow field produced by a Najafi-Golestanian swimmer is given by

u(x) := 1T

∫ T

0dt u(x) = 1

T

∫ T

0dt

∑

r

Gr(x)f r . (6.15)

Part of the difficulty in evaluating this average flow is that the position of the swimmer changes

with time, so that the distance |x − xr| is not a constant. Although the positions of each of

the spheres, xr(t), are in principle known from Eq. (6.12), the use of this directly does not

lead to any simple, or convenient expressions for the average flow. However, in the far field, at

distances large compared to the size of the swimmer, the magnitude of the oscillations in xr(t)

will be small compared to the distance to the point of observation. With this in mind we write

x− xr =(

x− y)

+(

y − xr)

=: r − δxr , (6.16)

where y is a fixed point used to represent the (average) position of the swimmer, and make use

of a multipole expansion of the stokeslet. In principle the choice of the point y is somewhat

arbitrary, however, for convenience we shall take it to be the position of the centre sphere at a

reference time, t = 0, corresponding to the start of the swimming stroke; y = x2(t = 0). The

stokeslet is then expanded as a formal power series in 1/r

Gr(x) =: 18πµ

∞∑

j=0

[

S(j)(r)](

δxr)⊗j

. (6.17)

The expansion coeffecients S(j) are formally defined by Eq. (6.17) from which it may readily


be shown that they are given by

S(j)αβ σ...τ (r) = (−)j

j! ∂σ . . . ∂τ

1r

(

δαβ + rαrβ)

. (6.18)

Inserting this multipole expansion for the stokeslets into Eq. (6.15), the far field average

flow becomes

uα(x) =∞∑

j=0

S(j)αβ σ...τ (r)nβnσ . . . nτ

∫ T

0

dtT

∑

r

(

δxr)j 1

8πµfr , (6.19)

since for the Najafi-Golestanian swimmer both the forces f r and the displacements δxr are

parallel to the swimming direction n. Eq. (6.19) is composed of two factors; the tensorial term

S(j)n⊗(j+1) which describes how the flow depends on position relative to the swimmer and how

it decays with distance, and an integral which captures how the details of the swimming stroke

determine the coefficient of each term in the multipole expansion. We can see immediately that

the leading j = 0 term vanishes on account of the total force generated by the swimmer being

zero. We shall calculate both of the next two terms; the j = 1 dipolar, and j = 2 quadrupolar,

contributions to the far field flow.

Dipolar flow

The most slowly decaying term is the 1/r2 dipolar contribution, so at large distances we can

expect this to provide an accurate description of the average flow field generated by most

swimmers. To calculate the integral in Eq. (6.19) we make use of the useful relations

δx1 = δx2 −(

D + ξR)

, and δx3 = δx2 +(

D + ξF)

, (6.20)

which allow us to write

∑

r

(

δxr)

f r =(

D + ξF)

f3 −(

D + ξR)

f1 ,

= 12

(

2D + ξR + ξF)(

f3 − f1)

+ 12

(

ξF − ξR)(

f3 + f1)

.

(6.21)


Inserting Eq. (6.9) for the forces we find that the relevant integral for the dipolar flow field is

a8T

∫ T

0dt

3(

2D + ξR + ξF + 3a2

)

∂t(

ξR + ξF)

+(

ξR − ξF)

∂t(

ξR − ξF)

+ a2

(

ξR − ξF)

[

1D+ξR

∂t(

2ξR + 4ξF)

− 1D+ξF

∂t(

4ξR + 2ξF)

− 12D+ξR+ξF

∂t(

ξR − ξF)

]

.

(6.22)

The first line is a total derivative and thus does not contribute to the final result. The remaining

integrals can be done using standard techniques to give

ωa2 sin(φ)8D

2ξF(

ξR − ξF cos(φ))(

DξR

)2[(

1 −( ξR

D

)2)−1/2

− 1]

+ ξF(

ξR + 2ξF cos(φ))(

DξR

)2[

1 −(

1 −( ξR

D

)2)1/2]

− 2ξR(

ξF − ξR cos(φ))(

DξF

)2[(

1 −( ξF

D

)2)−1/2

− 1]

− ξR(

ξF + 2ξR cos(φ))(

DξF

)2[

1 −(

1 −( ξF

D

)2)1/2]

− ξRξF(ξR+ξF)(ξR−ξF)8D2

(

2DΞ

)4[

2 −(

1 −(

Ξ2D

)2)1/2

−(

1 −(

Ξ2D

)2)−1/2]

,

(6.23)

where Ξ is again given by Eq. (6.14). Although this expression is quite lengthy and complicated

its most important feature is readily apparent: namely it vanishes identically if ξR = ξF. This,

of course, was to be expected. The Najafi-Golestanian swimmer with equal arm amplitudes is

self T-dual and thus, by Pooley’s corollary (§ 5.1.1), does not have a dipolar contribution to

its time averaged flow.

Quadrupolar flow

The quadrupolar term (j = 2) also makes an important contribution to the far field flow of

a Najafi-Golestanian swimmer. It decays more rapidly than the dipolar term, varying with

distance as 1/r3, however its amplitude turns out to be substantially larger so that over a

considerable range of intermediate distances from the swimmer, the time averaged flow field

may be well approximated as quadrupolar.

To evaluate the integral in Eq. (6.19) we again make use of the relations in Eq. (6.20) to


write

∑

r

(

δxr)2f r =

(

f3 + f1)

[

δx2(

ξF − ξR)

+(

D + 12 (ξF + ξR)

)2+ 1

4

(

ξF − ξR)2

]

+(

f3 − f1)(

D + 12(ξF + ξR)

)(

2δx2 + ξF − ξR)

.

(6.24)

Inserting the expressions for the forces from Eq. (6.9) and retaining only terms of quadratic

order in ξ the integral in Eq. (6.19) becomes

aD4T

∫ T

0dt

[

1+ 19a8D

]

(

ξF + ξR)

∂t(

ξF− ξR)

+3[

1+ 3a4D

]

(

2δx2 + ξF− ξR)

∂t(

ξF + ξR)

. (6.25)

We integrate the second term by parts and use Eq. (6.12) to substitute for ∂tδx2 to find that

Eq. (6.25) reduces to

17a2

32T

∫ T

0dt

(

ξF + ξR)

∂t(

ξF − ξR)

=17ωa2ξRξF sin(φ)

32. (6.26)

Far field flow

These two contributions, the dipolar and quadrupolar terms, provide a good description of

the far field properties of the average flow generated by the swimmer. Combining Eqs. (6.23)

and (6.26) with Eq. (6.19) we find that the average far field flow is

u(x) =21ωa2ξRξF

(

(ξR)2 − (ξF)2)

sin(φ)

256D3r2

[

3(

n · r)2 − 1

]

r

+17ωa2ξRξF sin(φ)

64r3

3(

n · r)[

5(

n · r)2 − 3

]

r−[

3(

n · r)2 − 1

]

n

+ o(1/r4) ,

(6.27)

where we have used the lowest order term in a series expansion in ξ/D for the amplitude of

the dipolar term. Comparing the amplitudes of these two contributions at points along the

swimming direction, we find that the flow field only becomes dipolar for distances

r

D&

136D2

21(ξR + ξF)|ξR − ξF| . (6.28)

This is a remarkable result, which shows that it is only appropriate to view the Najafi-

Golestanian swimmer as a simple force-dipole at distances of several tens of body lengths

from the swimmer. At closer distances the quadrupolar flow is more significant. However, it

should be cautioned that this result is in a sense a worst case scenario. The amplitude of the


dipolar term is required to vanish if either ξR = 0 or ξF = 0, since then the swimming stroke

is reciprocal, and also when ξR = ξF and the swimmer is self T-dual. This means that the

amplitude of the dipolar term must include a factor ξRξF(ξR− ξF). For sinusoidal strokes with

a single frequency, terms of cubic order in the oscillations all integrate to zero, leading to the

dipolar flow field scaling as ξ4, as in Eq. (6.27). However, this is not true for other swimming

strokes: in particular, for the original, four-stage swimming stroke [140] we have found that the

amplitude of the dipolar flow scales as ξ3, and that this dominates the far field time averaged

flow for distances [166]

r

D&

68D

29|ξR − ξF| , (6.29)

a rather more conservative result than Eq. (6.28).

The time averaged velocity field generated by the Najafi-Golestanian swimmer, using the

original four-stage swimming stroke, is shown in Fig. 6.2. It was obtained using a numerical

integration of the Oseen tensor equation, subject to the constraint that the swimmer is force

and torque-free. Details of this method are given in Reference [145]. Since the numerical

technique does not exploit a far field expansion of the stokeslets it enables the flow near the

swimmer to be determined as well, as illustrated in the inset. Close to the middle sphere

the fluid is clearly seen to move to the left; essentially the swimmer acts like a pump which

moves fluid from in front of it to behind it, leading to the swimmer moving to the right. The

analytic far field expansion, Eq. (6.27), accurately describes (to within 10%) the numerical flow

for distances further than around three swimmer lengths. The dashed circle demarcates the

estimated distance, Eq. (6.29), at which the far field flow crosses over from r−3 to r−2 dipolar

scaling.

It is interesting that two separate cases arise for the cross-over to a dipolar far field flow.

This is because the dipolar term is antisymmetric under ξR ↔ ξF while the quadrupolar term

is symmetric, giving distinct flow fields for the two cases ξR < ξF (Fig. 6.2(a)) and ξR > ξF

(Fig. 6.2(b)). As we shall show, this distinction results in significantly different long time

behaviour of interacting swimmers.


(a)

(b)

Figure 6.2: The flow field around a swimmer, averaged over one swimming cycle. The directionof swimming is from left to right and the flow is axisymmetric about the swimmer. Themagnitude of velocities are plotted on a logarithmic scale to enable both the near sphere (inset)and far field behaviour to be observed simultaneously. The parameters used were a = 0.1D,(a) ξR = 0.2D, ξF = 0.4D for contractile swimmers, and (b) ξR = 0.4D, ξF = 0.2D for extensileswimmers.

6.3 Swimming with friends 88

6.3 Swimming with friends

We now turn our attention to the hydrodynamics of more than one swimmer. That is, we wish

to ask what will be the influence on one swimmer of the presence of other swimmers nearby in

the fluid.

Our approach to calculating the hydrodynamics of a group of swimmers parallels our anal-

ysis of a single swimmer. In particular, the starting point is again the statement that linearity

of the Stokes equations allows the fluid flow to be written as a linear combination of the forces

acting on the fluid

u(x) =∑

A

∑

r

GrA(x)f rA . (6.30)

Here the subscript A labels the individual swimmers and the other notation is the same as in

Eq. (6.4). As for the single swimmer, we will consider that this expression provides a complete

solution if the forces f rA can be determined in terms of the prescribed swimming stroke. The

constraint that each swimmer is force-free still applies,∑

r frA = 0 ∀ A, however the consistency

relations on the fluid flow need to be modified to read (for each swimmer)

u(x2A) − u(x1

A) =: bRA =

(

∂tξRA

)

nA + [ΩA, (D + ξRA)nA] , (6.31)

u(x3A) − u(x2

A) =: bFA =

(

∂tξFA

)

nA + [ΩA, (D + ξFA)nA] , (6.32)

in order to account for the rotational motion of the swimmer. Here we employ an unconven-

tional bracket notation, [ , ], to denote the vector cross product. The angular velocity ΩA of

each swimmer is determined by imposing the further constraint that its motion impart no net

torque to the fluid∑

r

[

xrA − y , f rA]

= 0 . (6.33)

The point y about which the torque is measured is completely arbitrary because of the force-free

constraint. This remains true if the swimmer experiences a net external torque, but remains

force-free. However, if it is acted upon by an external force then y should be taken to be the

centre of mass.

As before, we use Eqs. (6.30) to (6.32) to set up a system of linear equations relating the

independent forces (f2A being eliminated via the force-free constraint) to the specified changing


shape of the swimmer. Schematically, these may be written as a matrix equation

∑

B

GABFB = BA , (6.34)

where

BA :=

b1A

b2A

, FA :=

f1A

f3A

, (6.35)

GAB :=

[G12B (x2

A) − G12B (x1

A)] [G32B (x2

A) − G32B (x1

A)]

[G12B (x3

A) − G12B (x2

A)] [G32B (x3

A) − G32B (x2

A)]

, (6.36)

and GrsB (x) = Gr

B(x) − GsB(x). The forces will be known if the matrix GAB can be inverted.

To perform the inversion we make use of the fact that the swimmers are in a dilute suspension

so that the interactions are all in the weak, far field regime. Consequently the elements GAAare much larger than GAB , A 6= B, which may be exploited in writing

FA = G−1AABA −

∑

B 6=AG−1AAGABFB . (6.37)

The final solution is then obtained by iteration. The first term represents the forces associated

with the swimming of a single isolated organism and are given as before by Eq. (6.9). The

second term gives the contribution to the forces due to interactions with all the other swimmers.

6.3.1 Interaction forces: expanding stokeslets

In treating the interactions between swimmers we need to consider objects of the form GrB(xsA),

stokeslets associated with spheres comprising swimmer B evaluated at the location of the

spheres of swimmer A. In the case of a dilute assembly of swimmers considered here it may

be assumed that the separation between swimmers is large compared to the size of any given

individual organism. Then, generalising Eq. (6.16), we introduce the decomposition

xrB − xsA =(

x2B(0) − x2

A(0))

+(

xrB − x2B(0)

)

−(

xsA − x2A(0)

)

,

=: rBA + δxrB − δxsA ,

(6.38)


and perform a multipole expansion of the stokeslets

GrB(xsA) =: 1

8πµ

∞∑

j=0

j∑

k=0

[

S(j,k)(rBA)](

δxsA)⊗k(

δxrB)⊗(j−k)

. (6.39)

This decomposition affords a convenient splitting of the interaction into an essentially geometric

piece, S(j,k)(rBA), dependent only on the relative position of the swimmers and terms dependent

on the details of the swimming motions. Again, the expansion coefficients S(j,k) are formally

defined by Eq. (6.39) from which it may be shown that they are given by

S(j,k)αβ σ...τ (r) = (−)k

k!(j−k)!∂σ . . . ∂τ

1r

(

δαβ + rαrβ)

. (6.40)

These tensors are fully symmetric in the j indices σ . . . τ as well as being symmetric in the

indices α, β. They scale with the separation between swimmers as r−(j+1)BA so that the terms

with the lowest values of j are expected to dominate the far field interactions. With this

expansion of the stokeslets the contribution to the forces coming from interactions is given by

(f INT)1A

(f INT)3A

α

= a4

∑

B 6=A

∞∑

j=2

j−1∑

k=1

S(j,k)αβ σ...τ ν...ρ(rBA)

−2(δx1A)⊗kσ...τ + (δx2

A)⊗kσ...τ + (δx3A)⊗kσ...τ

(δx1A)⊗kσ...τ + (δx2

A)⊗kσ...τ − 2(δx3A)⊗kσ...τ

∑

r

(

δxrB)⊗(j−k)ν...ρ

(

f rB)

β.

(6.41)

6.3.2 Swimmer rotation

Having determined both the single swimmer, Eq. (6.9), and interaction, Eq. (6.41), contribu-

tions to the forces, the sole remaining unknown is the angular velocity of the swimmer, ΩA.

This is obtained through imposing the constraint that the swimmer is torque-free, Eq. (6.33),

which we write as

0 =∑

r

[(

xrA − y)

, f rA]

,

=[(

x3A − x2

A

)

, f3A

]

−[(

x2A − x1

A

)

, f1A

]

,

=(

D + 12

(

ξRA + ξFA)

)

[

nA ,(

f3A − f1

A

)]

+ 12

(

ξFA − ξRA)[

nA ,(

f3A + f1

A

)]

.

(6.42)

It only proves necessary to balance the o(1) part of the single swimmer forces, Eq. (6.9), against

the contribution due to interactions, Eq. (6.41). There is no fundamental obstacle to retaining


the higher order terms from Eq. (6.9), however, since these lead to a substantial increase in

the length of formulae and play no essential role, they will be omitted in what follows. After

some straightforward manipulations the torque balance equation becomes

ǫαβγ

P nAβ nAδ [ΩA, ]γδ − a

4

∑

B 6=A

∞∑

j=2

j−1∑

k=1

S(j,k)γλ σ...τ ν...ρ(rBA) nAβA

kσ...τ C

j−kλ ν...ρ

= 0 , (6.43)

where we have defined

P := 6(

D + 12 (ξRA + ξFA)

)2+ 1

2

(

ξRA − ξFA

)2, (6.44)

Akσ...τ := 3D(

(δx3A)⊗kσ...τ − (δx1

A)⊗kσ...τ)

+ ξRA

(

−2(δx1A)⊗kσ...τ + (δx2

A)⊗kσ...τ + (δx3A)⊗kσ...τ

)

+ ξFA

(

−(δx1A)⊗kσ...τ − (δx2

A)⊗kσ...τ + 2(δx3A)⊗kσ...τ

)

,

(6.45)

Cj−kλ ν...ρ :=(

b1B)

λ

[

−2(δx1B)⊗(j−k)

ν...ρ + (δx2B)⊗(j−k)

ν...ρ + (δx3B)⊗(j−k)

ν...ρ

]

+(

b2B)

λ

[

−(δx1B)⊗(j−k)

ν...ρ − (δx2B)⊗(j−k)

ν...ρ + 2(δx3B)⊗(j−k)

ν...ρ

]

.

(6.46)

This equation is easily solved to give the angular velocity of each swimmer as

[ΩA, ]αβ = a4

∑

B 6=A

∞∑

j=2

j−1∑

k=1

(

δαγnAβ − δβγn

Aα

)

[

S(j,k)γλ σ...τ ν...ρ(rBA)

](

P−1Akσ...τ)

Cj−kλ ν...ρ . (6.47)

Our main interest is in the net effect that the interactions have over one complete swimming

stroke. Now, the orientation of each swimmer evolves according to the equation

dnAαdt

= [ΩA,nA]α . (6.48)

Technically, this is a set of Nswimmer coupled, non-linear, ordinary differential equations. We

make no attempt to solve it exactly, being satisfied with a perturbative approach. First we

solve for the motion of a single swimmer, which we then substitute into the right hand side of

Eq. (6.48). This means the right hand side may be considered as simply a function of t and

integrated directly. In this case a single swimmer does not rotate, so that where n appears on

the right hand side it may be considered just a constant vector. The net change in orientation


over a complete swimming stroke is thus

nAα (T ) − nAα = a4

∑

B 6=A

∞∑

j=2

j−1∑

k=1

(

δαβ − nAαnAβ

)

[

S(j,k)βγ σ...τ ν...ρ(rBA)

]

nBγ nAσ . . . n

Aτ n

Bν . . . n

Bρ

(

I(j,k))

,

(6.49)

where

I(j,k) :=

∫ T

0dt P−1Ak Cj−k . (6.50)

Again we emphasise that this expression decomposes the effect of the interactions into two

parts: the tensorial part involving S(j,k) which captures how the relative positions and orien-

tations of the swimmers influence their interactions, and the integral I(j,k) which captures the

details of the swimming stroke. In fact, the form of these integrals offers an important insight

into the nature of hydrodynamic interactions between swimmers. A short calculation reveals

that A1 = P so that all terms with k = 1 in Eq. (6.49) are independent of the swimming mo-

tion of swimmer A. We refer to these as passive interactions. They correspond to the rotation

that would be experienced by an inanimate object, or dead swimmer, drifting passively in the

flow field generated by the other swimmers. By contrast, the terms with k ≥ 2 depend on the

swimming motion of both swimmers. We call these active interactions. They represent the ad-

ditional rotation experienced by the swimmers because they are trying to swim simultaneously

and encode all of the information about the relative phase of the swimmers.

The most slowly decaying active term in the far field is the contribution with j = 3, k = 2,

which we now determine. A straightforward calculation gives

A2 = P[

2δx2A + 7

6

(

ξFA − ξRA)

]

− 13

(

ξFA − ξRA)3, (6.51)

C1 = 3D∂t(

ξFB + ξRB)

+ ∂t(

(ξFB)2 + ξFB ξRB + (ξRB)2

)

+ o(

a/D)

, (6.52)

so that the integral I(3,2) is given by

I(3,2) = 3πD2

ξRA[

ξRB sin(ηBA) + ξFB sin(ηBA − φB)]

− ξFA[

ξRB sin(ηBA + φA) + ξFB sin(ηBA + φA − φB)]

+ o(

ξ4)

.

(6.53)

Here, ηBA is the phase of swimmer B relative to swimmer A, i.e., if ξRA = ξRA sin(ωt) then

ξRB = ξRB sin(ωt + ηBA) and similarly for the front amplitudes. We remark in passing that this

amplitude does not vanish for φA = φB = π when the two swimmers are reciprocal. Instead we


find I(3,2) = (3πD/2)(ξRAξRB − ξFAξ

FB) sin(ηBA), so that, provided the swimmers are not in phase

or exactly out of phase, they can still interact hydrodynamically, despite each individually

performing a reciprocal motion. This insight will be discussed in more detail in Chapter 8.

Combining Eqs. (6.49) and (6.53), we find that the active contributions to the interactions

lead to a rotation of the swimmers that in the far field has the asymptotic form

∆nAactive ∼∑

B 6=A

−3aI(3,2)

8r4BA

[

1 + 2(nA · nB)2 − 5(nA · rBA)2 − 5(nB · rBA)2

− 20(nA · nB)(nA · rBA)(nB · rBA)

+ 35(nA · rBA)2(nB · rBA)2]

rBA − (nA · rBA)nA

.

(6.54)

For the passive interactions, the rotation can be determined using the time averaged flow field,

Eq. (6.27), from which we obtain

∆nApassive ∼∑

B 6=A

63πa2ξRBξFB

(

(ξRB)2 − (ξFB)2)

sin(φB)

128D3r3BA

[

(nA · rBA) + 2(nA · nB)(nB · rBA)

− 5(nA · rBA)(nB · rBA)2]


+∑

B 6=A

51πa2ξRBξFB sin(φB)

32r4BA

[

3(nA · nB) − 15(nA · rBA)(nB · rBA)

− 15(nA · nB)(nB · rBA)2 + 35(nA · rBA)(nB · rBA)3]


+[

(nA · rBA) + 2(nA · nB)(nB · rBA) − 5(nA · rBA)(nB · rBA)2]

×

nB − (nA · nB)nA

.

(6.55)

The most significant implication of this result is that the rotation generated by the active

interactions is substantially larger than that of the passive interactions for all reasonable sep-

arations. The passive terms scale as (a/D)2, while the active terms scale as a/D. Thus the

former are suppressed by an additional factor of the slenderness of the swimmer, which is al-

ways a small number for the Najafi-Golestanian swimmer. The important consequence of this

is that the relative phase, which enters only into the active terms, plays an important role in

the hydrodynamic interactions of microswimmers. This feature, which is central to our work, is

not captured by coarse-grained models of interacting swimmers that do not faithfully account

for the details of the swimming stroke.


6.3.3 Swimmer advection

In addition to a rotation of their direction of motion, the interactions also give rise to an

advection of each swimmer in the flow field produced by the others. This advection enters into

the determination of the translational motion of each swimmer described in § 6.2.1. Eq. (6.10)

still applies, although with the fluid velocity given by Eq. (6.30)

dx2A

dt= u(x2

A) =∑

s

GsA(x2

A)f sA +∑

B 6=A

∑

r

GrB(x2

A)f rB . (6.56)

The advection arises both directly, through the second∑

B 6=A term in Eq. (6.56), and indirectly,

from the fact that the forces f sA differ from their values for a single isolated swimmer on account

of the interactions. When these are combined we find that the advective contribution to u(x2A)

is, to leading order, given by

a12

∑

B 6=A

∞∑

j=1

j−1∑

k=0

[

S(j,k)(rBA)]

(

(δx1A)⊗k + (δx2

A)⊗k + (δx3A)⊗k

)

Cj−k . (6.57)

The equation for the translational motion, Eq. (6.56), is also a set of coupled, non-linear,

ordinary differential equations and hence we employ the same perturbative approach as for the

rotation, obtaining

[x2A]α(T ) − [x2

A]α = T nAα + a12

∑

B 6=A

∞∑

j=1

j−1∑

k=0

[

S(j,k)αβ σ...τ ν...ρ(rBA)

]

nBβ nAσ . . . n

Aτ n

Bν . . . n

Bρ

(

J (j,k))

,

(6.58)

where T is the single swimmer translational motion given by Eq. (6.13) and

J (j,k) :=

∫ T

0dt

(

(

δx1A

)k+

(

δx2A

)k+

(

δx3A

)k)

Cj−k . (6.59)

Again a distinction can be made between passive (k = 0) terms and active (k ≥ 1) terms.

As for the rotational interaction, the passive advection results from the same calculation that


gives the time averaged flow field, Eq. (6.27), and yields the result

∆x2A passive ∼

∑

B 6=A

21πa2ξRBξFB

(

(ξRB)2 − (ξFB)2)

sin(φB)

128D3r3BA

[

1 − 3(nB · rBA)2]

rBA

+∑

B 6=A

17πa2ξRBξFB sin(φB)

32r3BA

3(nB · rBA)[

5(nB · rBA)2 − 3]

rBA −[

3(nB · rBA)2 − 1]

nB

.

(6.60)

In addition to this passive advection there will be an active contribution, which we expect to

have a substantially larger amplitude, scaling as a/D instead of (a/D)2. Unlike the rotational

interaction, this does not come from the first non-trivial term, i.e., j = 2, k = 1. This is

because the sum of displacements δx1A+δx2

A+δx3A is itself o(a/D), since, as discussed following

Eq. (6.12), this quantity represents the centre of mass motion of the swimmer, which is only

non-zero because of hydrodynamic interactions between the spheres. Thus the leading active

interaction comes from the term j = 3, k = 2, for which we find

J (3,2) = −6πD2

ξRA[

ξRB sin(ηBA) + ξFB sin(ηBA − φB)]

+ ξFA[

ξRB sin(ηBA + φA) + ξFB sin(ηBA + φA − φB)]

,

(6.61)

and the active advection is then given by

∆x2A active ∼

∑

B 6=A

−aJ (3,2)

8r4BA

[

1 + 2(nA · nB)2 − 5(nA · rBA)2 − 5(nB · rBA)2

− 20(nA · nB)(nA · rBA)(nB · rBA) + 35(nA · rBA)2(nB · rBA)2]

rBA

+ 2[


nA

.

(6.62)

6.3.4 An improved near field calculation

A drawback of the approach we have just described is that it is based on a far field analysis,

where the separation between swimmers is large compared to the size of an individual, D/r ≪ 1.

Since the Oseen tensor approach allows us to determine the hydrodynamic interactions between

two spheres of the same swimmer when they are a distance D apart (for the determination of

the single swimmer motion), it should also allow us to determine the interactions between two

spheres of different swimmers when the separation between them is also o(D). Clearly this

cannot be done on the basis of a multipole expansion in powers of D/r, since for r = o(D) this


will be, at best, very slowly convergent. At these close separations we can replace Eq. (6.38)

for the relative position of the two spheres with

xrB − xsA =(

xrB(0) − xsA(0))

+(

xrB − xrB(0))

−(

xsA − xsA(0))

,

=: r(r,s)BA + δxrB − δxsA .

(6.63)

If we consider that the deviation of each sphere from its “mean” position is small compared to

the separation between them, ξ/D ≪ 1, then we can still make use of an expansion of stokeslets

to determine the interactions in the same way as before. The price that we pay for this is that

the tensors S(j,k) appearing in Eq. (6.39) are no longer independent of the pair of spheres (r, s),

so that each pair needs to be considered individually. This leads to a substantial increase in

the length of our formulae. For example, Eq. (6.47) becomes

[ΩA, ]αβ = a4P

−1(

δαγnAβ − δβγn

Aα

)

∑

B 6=A

∞∑

j=0

j∑

k=0

nBλ nAσ . . . n

Aτ n

Bν . . . n

Bρ

[

S(j,k)γλ σ...τ ν...ρ(r

(1,1)BA )

] (

δx1A

)k(δx1

B

)j−k[

(

3D + 2ξRA + ξFA)

∂t(

2ξRB + ξFB)

]

+[


(1,2)BA )

] (

δx2A

)k(δx1

B

)j−k[

(

ξFA − ξRA)

∂t(

2ξRB + ξFB)

]

+[


(1,3)BA )

] (

δx3A

)k(δx1

B

)j−k[

−(

3D + ξRA + 2ξFA)

∂t(

2ξRB + ξFB)

]

+[


(2,1)BA )

] (

δx1A

)k(δx2

B

)j−k[

(

3D + 2ξRA + ξFA)

∂t(

ξFB − ξRB)

]

+[


(2,2)BA )

] (

δx2A

)k(δx2

B

)j−k[

(

ξFA − ξRA)

∂t(

ξFB − ξRB)

]

+[


(2,3)BA )

] (

δx3A

)k(δx2

B

)j−k[

−(

3D + ξRA + 2ξFA)

∂t(

ξFB − ξRB)

]

+[


(3,1)BA )

] (

δx1A

)k(δx3

B

)j−k[−(

3D + 2ξRA + ξFA)

∂t(

ξRB + 2ξFB)

]

+[


(3,2)BA )

] (

δx2A

)k(δx3

B

)j−k[

−(

ξFA − ξRA)

∂t(

ξRB + 2ξFB)

]

+[


(3,3)BA )

] (

δx3A

)k(δx3

B

)j−k[

(

3D + ξRA + 2ξFA)

∂t(

ξRB + 2ξFB)

]

.

(6.64)

The leading order contribution to the active interaction comes from the j = 0, k = 0 and

j = 1, k = 1 terms and thus can be thought of as a collection of stokeslets, S(0,0), and stokes

doublets, S(1,1) [171]. The integrals over a complete swimming stroke of these terms are straight-

forward, but the number of them makes their evaluation tedious and we shall not quote the

results here. The advection can be treated in a similar fashion, but again the results will not

be quoted explicitly due to the length of the final expressions.

6.4 Trajectories of two parallel swimmers 97

Comparison between near and far field approaches

A comparison of the results of the near and far field calculations is presented in Fig. 6.3 for

two parallel, coplanar swimmers. First, in Fig. 6.3(a) we show how the rotation angle ∆θ

depends on the separation between the swimmers in the near field calculation. This is found

to asymptote onto the predicted far field scaling of r−4BA for distances of greater than about

10D, but to become less singular at closer distances. In Figs. 6.3(b) and (c) we show how ∆θ

varies as one swimmer is moved in a circle around the other, keeping a fixed separation between

them. At a separation of 5D (Fig. 6.3(b)) there are noticeable differences between the near

and far field calculations, although the qualitative features are the same. These discrepancies

are substantially reduced when the separation is increased to 10D (Fig. 6.3(c)), indicating that

the far field description is sufficient.

The simulation parameters used for this comparison were a = 0.05D, ξR = ξF = 0.2D,φA =

φB = π/2 and ηBA = 0.

6.4 Trajectories of two parallel swimmers

We conclude this Chapter by applying the results of § 6.3 to a simple problem involving the

hydrodynamic interactions between just two swimmers. The two swimmers are initially oriented

along parallel directions and we consider how their subsequent long time trajectories change

as one swimmer is moved in a circle around the other one, keeping them a fixed separation of

5D apart.

We consider first self T-dual swimmers (ξR = ξF) swimming in phase, before looking at

how the behaviour changes when these conditions are relaxed. The results are summarised in

Fig. 6.4(a). Swimmer A initially lies at the origin (as shown in the figure) and swimmer B is

aligned parallel and displaced to the corresponding point in the diagram. The figure is divided

into different regions, labelled A, R, O, and P, corresponding to the four different types of long

term behaviour that were observed; attractive, repulsive, oscillatory and parallel trajectories.

Typical trajectories for each of these types of behaviour are also shown in Fig. 6.4(i-iv). For

accuracy, the trajectories were obtained numerically [145], although the results which follow

from iterating the analytic formulae, Eqs. (6.54) and (6.62), are in close agreement. All scales

are in units of D, the swimmer arm length. The parameters chosen were a = 0.1D and

ξ = 0.3D, giving a displacement per cycle of T ≈ (7π/12)aξ2/D2 = 0.016D. Thus, since the


Figure 6.3: Comparison of the near and far field calculations of the hydrodynamics of Najafi-Golestanian swimmers. (a) The dependence of the rotation angle on separation for two parallelswimmers, measured in the direction perpendicular to their orientation. (b) and (c) Variationof the rotation angle with the relative position of the two swimmers at a fixed separation of(b) rBA = 5D and (c) rBA = 10D. In both cases the solid blue line represents the near fieldcalculation and the dashed black line the far field calculation.


length scale of the swimming motion is typically ∼50D, the trajectories evolve over timescales

3 or 4 orders of magnitude larger than that of the individual swimming stroke. The lines

terminate when the swimmers get too close, as here the Oseen tensor approximation breaks

down.

In addition to simple attractive or repulsive behaviour the swimmers also exhibit intriguing

oscillatory trajectories which may either be of small amplitude about separated parallel swim-

ming paths, as in region P, or of larger amplitude with the two swimmers’ trajectories crossing

each other, as in region O. These oscillations take place over the course of ∼10, 000 swimming

cycles, although this period varies strongly with the separation of the two swimmers.

The type of trajectories observed depends on the relative phase of the swimmers. Fig. 6.4(b)

shows the analogous diagram to Fig. 6.4(a), but with the two swimmers π out of phase. The

trajectories change because the amplitude of the active rotation changes sign. The attractive

region is replaced by a repulsive one. The oscillatory regime now produces oscillating parallel

paths, an example of which is shown in Fig. 6.4(iv). Here the two swimmers follow parallel

paths inclined at a small angle to the direction in which they would move in the absence

of interactions. Both trajectories exhibit small amplitude, long period oscillations about the

swimming direction; the oscillations of the two swimmers are exactly π out of phase, and the

amplitude of the oscillations vanishes when the angle between them is ∼30o, corresponding to

one of the zeroes of ∆nactive, Eq. (6.54).

Fig. 6.4(c) shows the intermediate case when the relative phase of the two swimmers is

ηBA = π/2. This case is noteworthy as the amplitude for the lowest order part of the active

rotation, Eq. (6.53), vanishes identically. Nonetheless, it is clear that the active terms still

dominate over the passive interactions since the behaviour is different for swimmers placed

in front or behind the central swimmer. This positional change is equivalent to changing the

relative phase between the swimmers from π/2 to 3π/2 and thus can only be important if the

active terms are dominating the interaction.

The oscillations of Fig. 6.4(iii) are perfectly periodic and repeat indefinitely. This is a di-

rect consequence of the self T-duality of the Najafi-Golestanian swimmer when the amplitudes

of the two arms are equal. For swimmers that are not self T-dual, the oscillating states are

no longer indefinitely stable. In particular, Fig. 6.4(v) shows the effect of making the front

swimming amplitude, ξF, smaller and the rear amplitude, ξR, larger. Now the amplitude of

the oscillations becomes successively larger until they can no longer be sustained and, finally,


the two swimmers follow diverging trajectories. The opposite case, when ξF > ξR, is shown in

Fig. 6.4(vi). The amplitude of the oscillations decays and the swimmers move closer together,

with the lines terminating when the Oseen tensor approximation breaks down as a result of

the swimmers getting too close.

In this Chapter we have described the interactions between two simple model linked-sphere

swimmers. These interactions are not of the naively assumed dipolar form, nor do they originate

from the average flow field generated by any given individual. Instead they arise from an active

term, coming from the simultaneous efforts of both swimmers to swim. This results not only

in a more rapid, r−4BA, decay with separation than would occur for dipolar interactions, but also

in a sensitive dependence of the interactions to the relative phase between the two swimmers,

a feature recognised as important by Taylor [114], but not explicitly incorporated into many

minimal swimmer models. In the next Chapter we shall investigate the form of the interactions

between two swimmers further by considering swimmer scattering, and will find again that the

relative phase plays an important role.


(i)

(ii)

(iii)

O O

R RP A

R RPAP

P

RR

O

APR

P

O

P

P

O R

AP

(iv)

R OO

P

RO

P

O

(vi)

R RP A

R RPAP

P

OCOC

0 10 20 300

1

2

3

4

5

0 20 40 60−10

−5

0

5

10

0 20 40 60 80 100−3

−2

−1

0

1

2

0 50 100 150−1

0

1

2

3

0 100 200 300 400−20

−10

0

10

20

0 100 200 300−6

−4

−2

0

2

(b)

(c) (d)

(a)

OC

(i) (ii)

(iv)(iii)

(vi)

P

RA

O

OE(v)

Figure 6.4: Long time behaviour of two Najafi-Golestanian swimmers, (a) in phase, (b) π outof phase, (c) ηBA = π/2, and (d) in phase, but with ξF > ξR. (i)–(vi) illustrate a selection oftrajectories representing attractive (A), repulsive (R), oscillatory (O), parallel (P), expandingoscillatory (OE), and contracting oscillatory (OC) behaviour, respectively.

Chapter 7

Swimmer scattering

Consider two swimmers that are approaching each other in the fluid. As they approach the

hydrodynamic interactions between them will cause them both to rotate and subsequently to

separate along new trajectories. This classic scenario describes a scattering event and provides

the focus of our attention for this Chapter.

7.1 Geometry of scattering

In a generic scattering event the two swimmers, A andB, have instantaneous positions xA(t),xB(t)

and swimming directions nA(t),nB(t). We use these positions and orientations to describe the

configuration of the swimmers via a simple geometric construction, shown in Fig. 7.1. The po-

sition and orientation of each swimmer generates a straight line. Provided the two swimming

directions are not parallel, there will be unique points, lA and lB, at which they are closest to

each other. We denote by sA, sB the distances of each swimmer from these points of closest

approach, taken positive if l is ahead of the swimmer and negative otherwise. The difference

between these two distances, b := sA − sB, is called the impact parameter and measures how

closely the two swimmers would approach each other in the absence of any interactions. The

minimum distance between the two lines, l := |lA − lB |, is called the lift parameter and the

angle between the two swimmers is called the angle of incidence, α := arccos(nA · nB). The

limiting values of these quantities are used to define the initial and final states

(

α±, b±, l±)

= limt→±∞

(

α(t), b(t), l(t))

. (7.1)

102

7.2 Scattering trajectories 103

Figure 7.1: Schematic diagram of the geometry of a scattering event: see text for details. Forease of illustration we show a planar event, i.e., l = 0.

Finally, the change in orientation of an individual swimmer as a result of the scattering process

is described by the scattering angle

θ = arccos(

n(t → −∞) · n(t→ +∞))

. (7.2)

7.2 Scattering trajectories

The scattering trajectories of two Najafi-Golestanian swimmers can be determined using the

results presented in Chapter 6. We briefly describe the general features, for planar scattering

geometries, l = 0, which set the scene for this Chapter. Three generic types of scattering event

are observed: the swimmers can collide, by which we mean they approach so closely that the

Oseen tensor approximation is no longer appropriate, they can exchange trajectories, or they

can turn in the same direction. Exemplary trajectories for the latter two types of events are

shown in Fig. 7.2. In the turn event the impact parameter is relatively large and swimmer B

is able to pass directly in front of swimmer A, causing both of them to rotate in the clockwise

sense. For the exchange event, the impact parameter is slightly smaller, and in this case the

swimmers do not cross, but rotate in opposite directions, essentially swapping places. The

remarkable feature of both events is that the angle between the swimmers is unchanged by the

scattering.

7.3 T-duality and swimmer scattering 104

Figure 7.2: Trajectories of two Najafi-Golestanian swimmers during planar scattering. (I) aturn event, and (II) an exchange event.

7.3 T-duality and swimmer scattering

In this Section we shall describe the consequences of the time reversal invariance of the Stokes

equations for the hydrodynamic scattering of two low Reynolds number swimmers, and argue

that this provides the explanation for the preservation of angle described in § 7.2.

Hydrodynamic scattering may be viewed as providing a map from the initial state (α−, b−, l−)

to the final state (α+, b+, l+). The differences between these two states, defined by the func-

tions δα := α+ − α−, δb := b+ − b− and δl := l+ − l−, describe the tendancy for the swimmers

to align (δα < 0) or cluster (δb < 0, δl < 0) via hydrodynamic interactions. Viewing the entire

process backwards in time corresponds to the hydrodynamic scattering of the T-dual swimmers

(B, A) taking the initial state (α+, b+, l+) into the final state (α−, b−, l−), thereby establishing

an isomorphism between the scattering of an arbitrary pair of swimmers and the scattering of

their T-duals. In particular the functions δ(B,A) are simply related to the functions δ(A,B)

δ(B,A)(α+, b+, l+) = −δ(A,B)(α−, b−, l−) . (7.3)

In the case of a pair of mutually T-dual swimmers (B = A; A = B) this is sufficient to show

that (α+, b+, l+) ≡ (α−, b−, l−).

We motivate this result using symmetry arguments. During any scattering event the quan-

tity sA+sB changes from being large and positive to being large and negative. Since it does this

continuously it must pass through zero, which we use to define the time t = 0. The separation


Figure 7.3: Time reversal transformations and swimmer scattering. At t = 0 reversing thedirection of time and performing a π rotation about the axis indicated returns an identicalconfiguration if the pair are mutually T-dual. Top row: a turn event. Bottom row: anexchange event.

between the two swimmers is given by

rBA = 12

(

sA + sB) (

nA − nB)

+ 12

(

sA − sB) (

nA + nB)

+(

lB − lA)

, (7.4)

and is orthogonal to the direction nA − nB at t = 0. At this instant reversing the direction

of time, followed by a π rotation about an axis parallel to nA − nB and passing through the

point mid-way between the two swimmers leads to a configuration where B, A have the same

positions and orientations as A,B, respectively. As illustrated in Fig. 7.3, for mutually T-dual

swimmers this returns the same configuration we started with. It follows that A’s outgoing

trajectory for t > 0 will be given by B’s ingoing trajectory for t < 0 (with the direction of

time reversed) and vice-versa, from which we conclude that the initial and final states are

the same. In addition, for planar scattering geometries, this construction implies that the

swimmers rotate in the same direction and with equal scattering angles, θA = θB. These are

the turn events, of which an example was shown in Fig. 7.2(I).

An exception to this scenario occurs if the swimmers ever become exactly parallel, nA = nB .

However, taking t = 0 at this instant and choosing the rotation axis to be parallel to nA× rBA

leads to the same conclusion (Fig. 7.3). This time, since the two swimmers rotate in opposite

directions, the constraint that α+ = α− can only be met if the scattering angles take the values

θ = ±α− independent of b−. In such an event A will rotate so that its outgoing trajectory is


parallel to B’s ingoing trajectory and vice-versa. These are the exchange events, an example

of which was shown in Fig. 7.2(II). Since we expect θ → 0 as b− → ∞, exchange events can

only occur for sufficiently small values of b−. Finally, we comment that, since for purely planar

scattering there is no way to cross smoothly between these two cases, they are necessarily

separated by some form of discontinuous behaviour. We speculate that the swimmers enter a

trapped, or bound, state from which they are unable to escape.

There are a number of subtleties in the foregoing observations. To exactly interchange

the swimmers as described, the time t = 0 must coincide with particular instants during the

swimming cycle, otherwise, although the positions and orientations of the swimmers will be

the same after the interchange, the stages they are at during their swimming strokes will not.

We have not been able to show generally that the time t = 0 does indeed coincide with one of

these instances, although for this not to be the case would imply rather peculiar properties for

the functions δ(A,A). Also, since our discussion has mentioned only the swimming direction,

n, it has been restricted to swimmers that are axisymmetric, requiring only this vector to

completely specify their orientation.

Moreover, we have defined the time t = 0 through the condition that the vector nA − nB

is orthogonal to the relative position vector rBA of the two swimmers, but we have not defined

what this means. In particular, what is the relative position of the two swimmers? Is it the

separation between the two centre spheres? Or between the two leading spheres (the ‘head’

of the swimmer)? In fact, it does not matter which point (i.e., which sphere) is chosen to

represent the position of a given swimmer, a consequence of the insignificance of inertia at

low Reynolds number [119]. Different choices correspond to a change of ‘gauge’ and invariance

of the single swimmer motion under gauge transformations leads to the beautiful gauge field

theory of swimming introduced by Shapere and Wilczek [118–120].

But different choices do matter here. If the vector nA − nB is orthogonal to the relative

position vector of the two centre spheres it will not simultaneously be orthogonal to the relative

position vector of the two ‘head’ spheres. Nonetheless, a unique definition of t = 0 is still

possible. nA − nB cannot be simultaneously orthogonal to the relative positive vectors of all

pairs of points on the two swimmers, but it can be simultaneously orthongal to all pairs of

conjugate points. Conjugate points are defined between a swimmer and its T-dual such that

they correspond to the same material point on the two swimmers. This is most easily illustrated

by a figure, as we have done in Fig. 7.4. The condition that the relative position vectors of


Figure 7.4: The relationship between a swimmer and its T-dual naturally introduces the notionof conjugate points. The foremost part of a swimmer is conjugate to the rearmost part of itsT-dual and vice-versa.

all pairs of conjugate points are simultaneously orthogonal to the vector nA − nB is sufficient

to guarantee that, after the interchange described above, B has the same position, orientation

and is also at the same stage through the swimming stroke, as A before the interchange.

In numerical tests using both the Najafi-Golestanian model and a model of snake-like swim-

mers, the swimmers do indeed reach t = 0 at a suitable stage of their stroke. This is shown in

Fig. 7.5 for both turn and exchange type events with the Najafi-Golestanian swimmer. Under

T-duality, the rear arm of swimmer A becomes the front arm of swimmer A = B. Thus, for the

interchange of the two swimmers to be exact it must occur at a stage through the swimming

stroke where the rear arm of A has the same extension as the front arm of B. Parameterising

the swimming stroke by a variable τ ∈ [0, 2π), we see that this implies the condition

sin(τ) = sin(τ + ηBA − φ) , (7.5)

which is satisfied for τ = ((2m+1)π+φ− ηBA)/2, with m = 0, 1. In the simulations described

here, the swimmers were in phase ηBA = 0 and the phase lag between front and rear arm

oscillations was φ = π/2. For these values the interchange is exact only if τ = 3π/4 or

τ = 7π/4. As can be seen in Fig. 7.5, for the turn event the vector nA − nB is orthogonal to

rBA for all three pairs of conjugate points simultaneously at a stage corresponding exactly to

τ = 7π/4, while for the exchange event the two swimmers are parallel at a stage corresponding

exactly to τ = 3π/4.

7.4 Scattering of the Najafi-Golestanian swimmer 108

Figure 7.5: Numerical identification of the time t = 0 in the scattering of a pair of mutuallyT-dual swimmers, verifying the exact interchange. The black arrows indicate the start of swim-ming stroke of swimmer A, each swimming stroke corresponding to 80 simulation timesteps.For the turn event the three sets of data correspond to the three choices of conjugate points tomeasure the relative position rBA. These are: the tail sphere of A and head sphere of B (red),the two centre spheres (blue), and the head sphere of A and the tail sphere of B (cyan). I amgrateful to Vic Putz for providing these figures.

7.4 Scattering of the Najafi-Golestanian swimmer

These general symmetry considerations provide a framework for what can be expected in

two body swimmer scattering. In the remainder of this Chapter we illustrate and extend

the results by describing in more detail the planar hydrodynamic scattering of two Najafi-

Golestanian swimmers. We consider first two identical swimmers that are in phase, ηBA = 0,

and have equal arm amplitudes, ξR = ξF, for which the swimming stroke is self T-dual. For all

trajectories (α+, b+) = (α−, b−) in accordance with the symmetry arguments we have presented

above. The type of scattering event (exchange or turn) that occurs is shown as a function of

the two initial conditions α−, b− in Fig. 7.6(a), together with a detailed cut showing how the

scattering angle θ varies with b− for a fixed value of α− = 30o (Fig. 7.6(b)). There is a wide

range of initial conditions at small values of b− for which the scattering is of the exchange

type and θ = ±α−. At larger values of b− the scattering is always of the turn type with the

scattering angle decaying to zero as b− → ∞. In the region labelled Collide the swimmers

approach so closely that the Oseen tensor description of the hydrodynamics is no longer valid

and we are unable to determine what happens during the scattering. In our simulations we

took this to occur when the minimum separation between any two spheres became less than

10a = 0.5D.

The behaviour is similar for different values of the relative phase between the swimmers. In


Figure 7.6: Hydrodynamic scattering of two identical, self T-dual Najafi-Golestanian swimmers.(a) and (c) The type of scattering observed for different values of the initial conitions α−, b−when the swimmers are in phase, ηBA = 0, and exactly out of phase, ηBA = π, respectively. (b)and (d) Dependence of the scattering angle on b− for a fixed value of α− = 30o, correspondingto the dashed lines in (a) and (c), respectively.

Fig. 7.6(c) and (d) we show the results for two swimmers that are exactly out of phase, ηBA = π.

The same types of scattering are found, but with the boundaries between the different regions

shifted to smaller values of b−. An interesting feature is that the scattering angle in the turn

regime does not decay to zero monotonically with increasing b− as is the case for swimmers that

are in phase. At large values of b− the scattering angle is negative for ηBA = 0 and positive for

ηBA = π, reflecting the change in sign of the amplitude of the far field rotational interaction as

given by Eq. (6.53). If the far field approximation to the hydrodynamics provided a sufficient

description of the interactions we might expect the scattering angle to be positive for all b−

when the swimmers are exactly out of phase, i.e., simply the opposite sign to when they are in

phase. That this is not the case reflects the need to capture the interactions more accurately

when the separation between the swimmers is comparable to their size.

We now outline how the properties of swimmer scattering change when the two swimmers

are not mutually T-dual and our preceeding symmetry arguments no longer apply. In Fig. 7.7(a)

we show the change in alignment δα as a function of b− for both a pair of identical extensile

swimmers, with (ξR, ξF) = (0.3D, 0.1D), and a pair of identical contractile swimmers, with


(ξR, ξF) = (0.1D, 0.3D), with a relative phase of ηBA = π/2 in both cases. Not only is δα non-

zero, it is generically large; several tens of degrees in many cases. For this choice of relative

phase we see a predominant tendancy for extensile swimmers to increase the angle between the

two swimmers, i.e., δα > 0, while contractile swimmers predominantly show hydrodynamically

induced alignment, δα < 0.

However, the relative phase is found to play an important role here. We show in Fig. 7.7(b)

the change in alignment δα as a function of the relative phase for fixed α− = 30o, b− = 4D and

the same two pairs of swimmers. Several features are noteworthy. The most striking feature is

the appearance of an extended region where δα = −30o for the pair of contractile swimmers.

This represents the formation of a bound state in which the two swimmers are exactly aligned

one behind the other. An example of swimmer trajectories during the formation of this bound

state is shown in Fig. 7.7(d). A smaller region exhibiting the same behaviour can also be seen

in Fig. 7.7(a).

As the relative phase is varied, δα is as often positive as it is negative, for both extensile

and contractile pairs of swimmers. Thus there is not a simple, direct relationship between the

change in angle between the two swimmers and their being contractile or extensile. However,

the behaviour is asymmetric in the sense that the magnitude of δα is larger for 0 < ηBA < π

than for π < ηBA < 2π in both cases. Moreover, there are two values of the relative phase,

ηBA = 0 and ηBA = π, for which δα = 0. Thus we find an intriguing result: when the swimmers

are exactly in phase, or exactly out of phase, hydrodynamic scattering does not change the

angle between them, even if they are not mutually T-dual. It is tempting to speculate that

this might also follow from symmetry considerations, or that it might apply to swimmers other

than the Najafi-Golestanian model considered here.

On either side of the bound state regions in both Fig. 7.7(a) and (b), δα takes a value

somewhat larger than −30o, indicating that δα is discontinous at the transition from scattering

to bound state formation. This qualitative observation is supported by the behaviour when

the transition is approached from a different direction, that of increasing the difference in am-

plitudes ξF − ξR at a fixed value of b− = 4D as shown in Fig. 7.7(c). We vary this difference

in amplitudes whilst holding the product ξRξF fixed to maintain an approximately constant

swimming speed, Eq. (6.13), [140, 143, 166]. As ξF − ξR is increased from zero δα decreases

smoothly until it reaches the value 0.16D where there is an abrupt transition to the bound

state, accompanied by a substantial discontinuity in δα.


Figure 7.7: Hydrodynamic scattering of Najafi-Golestanian swimmers that are not mutuallyT-dual. The change in angle between two swimmers as a function of (a) the impact parameterfor a fixed relative of phase ηBA = π/2, and (b) their relative phase at a fixed value of theimpact parameter, b− = 4D. In both cases the red circles correspond to a pair of identicalextensile swimmers with ξR = 0.3D, ξF = 0.1D and the blue crosses correspond to a pair ofidentical contractile swimmers with ξR = 0.1D, ξF = 0.3D. In (c) we show how the change inangle between the swimmers depends on the amplitude asymmetry ξF − ξR for b− = 4D andηBA = π/2. In (d) we show an exemplary scattering trajectory leading to the formation of abound state for ξF − ξR = 0.17D and other parameters as in (c).

7.5 Discussion 112

7.5 Discussion

We have described the constraints imposed on the hydrodynamic scattering of two swimmers by

the time reversal invariance of the Stokes equations. The most striking observation concerns T-

dual swimmers, which have strokes that map onto each other under time reversal: for scattering

events involving two such swimmers the angle and impact parameter between their trajectories

are the same before and after the collision. For swimmers unrelated by T-duality we show

numerically that scattering is complex, with changes in the angle between the two swimmers

of several tens of degrees, and allowing the possibility of forming bound states. Experiments

on biological or fabricated microswimmers should show these striking differences between pairs

of mutually T-dual and symmetry unrelated swimmers.

In future work it would be particularly interesting to extend our ideas to more than two

swimmers. In suspensions of self T-dual swimmers the evolution of the system may be con-

strained to look the same both forwards and backwards in time, at least in a statistical sense,

and it may be possible to exploit this to obtain exact results for their non-equilibrium dynam-

ics. Furthermore, these microscopic symmetries of the interactions between individuals should

also appear in the coarse-grained hydrodynamic equations of continuum theories. It will be

important to determine in what way self T-duality enters into the continuum equations and

how it effects the resulting macroscopic behaviour.

The description of the scattering of two swimmers described in this Chapter is not yet a

complete one and there are many aspects that merit further study. Determining the limiting

behaviour of the swimmers during the discontinuous cross-over from turn to exchange type

events is one such unresolved problem, but this is beyond the validity of the Oseen tensor

approach. Another is the status of non-axisymmetric swimmers; it is not yet clear whether

or not they exhibit the same exact recovery of the initial state in the long time limit as we

described for axisymmetric swimmers. It is tempting to speculate that they do, but this still

has to be shown, either generally or at least for some specific cases. And this brings us to our

final comment. A proof of the remarkable scattering behaviour described in this Chapter is

still needed; we have only been able to provide part of it. It is both interesting and important

to complete it.

Chapter 8

Dumb-bell swimmers

In the low Reynolds number world of microorganisms, reciprocal strokes do not lead to swim-

ming [116, 117, 169]. Organisms that perform such reciprocal motions, referred to as apolar

swimmers or “shakers” [109], are able to generate fluid flow but unable to generate their own

motility. However, as pointed out by Koiller et al [172], although this is true for a single ap-

olar swimmer, a collection of such swimmers can swim. If their swimming strokes are not in

phase, the motion taken as a whole is not reciprocal in time, and motility is possible. The

physical mechanism which turns the motion of an apolar swimmer into a net displacement is

the hydrodynamic forces between the swimmers.

In this Chapter we study the simplest model of an apolar swimmer, oscillating dumb-bells.

Following the approach we developed in Chapter 6, we give an analytic solution for the motion

of a pair of dumb-bells, in the Oseen tensor limit and valid for large separations. By numerically

iterating these results we are able to gain insight into how the orbits of two swimmers depend

on their relative positions, separations, and phases.

The motility of an apolar swimmer is inherently collective: a single dumb-bell does not

swim, but more than one can. It is natural, therefore, to investigate how this collective lo-

comotion develops as the number of dumb-bells is increased. We do this by first considering

simple systems, principally linear chains and regular arrays, and describe how the resulting

motion can be understood in terms of the pairwise interactions between individuals.

Finally we describe the motion of a suspension of randomly placed and randomly oriented

dumb-bells. Since each dumb-bell is intrinsically apolar, a suspension of them provides a

minimal microscopic model of an active apolar fluid. These active fluids have received a lot of

interest in the past decade [98,99,108,109,173–175], both as examples of driven non-equilibrium

113

8.1 Hydrodynamic interactions between oscillating dumb-bells 114

systems and because of their close analogy to active polar fluids [98,99,176–178]. The properties

of such systems have been formulated primarily at the continuum level [98, 99], on the basis

of symmetry and conservation laws, but have also been studied numerically for suspensions of

individuals interacting through simple modelling rules [174].

8.1 Hydrodynamic interactions between oscillating dumb-bells

We first describe the motility of oscillating dumb-bells at zero Reynolds number. Each dumb-

bell comprises two spheres, of radius a, joined by a thin, rigid rod, whose length varies sinu-

soidally as D + ξ sin(ωt). As in other linked sphere models the rod is treated as a phantom

linker and its hydrodynamic effect neglected. We stress the importance of the relative phase of

the two dumb-bells: if they oscillate in phase or π out of phase then their combined movements

remain reciprocal and the Scallop theorem prevents any net motion [117]. However, for other

values of the relative phase, motion will occur. To show this we use the Oseen tensor formu-

lation of hydrodynamics, valid in the limit of zero Reynolds number [170], with an analysis

similar to Chapter 6. We present anaytic calculations, valid for large separations, and numer-

ical results based on a more detailed near field calculation like that described in § 6.3.4, valid

for separations down to of order the size of a dumb-bell.

As described in Chapter 6, linearity of the Stokes equations allows the fluid velocity to

be written as a linear combination of the forces acting on the fluid due to the motion of the

dumb-bells

u(x) =∑

A

[

G1A(x)f1

A + G2A(x)f2

A

]

. (8.1)

Here the subscript A labels the dumb-bells, the superscripts 1, 2 label the spheres of an in-

dividual dumb-bell and the Green function is again taken to be the Oseen tensor, Eq. (6.5),

assuming a/D ≪ 1. The motion of the dumb-bells through the fluid is determined, as before,

by three ingredients; consistency of the fluid flow with the change in shape, i.e., the change in

rod length, the constraint that each dumb-bell is force-free, f1A + f2

A = 0, and the constraint

that each dumb-bell is torque-free, [(x2A − x1

A) , f1A] = 0.

8.1 Hydrodynamic interactions between oscillating dumb-bells 115

A short calculation then leads to an expression for the forces

f1A = −3πµa

ξω cos(ωt)(

1 + 32

aD+ξ sin(ωt)

)

nA +(

D + ξ sin(ωt) + 3a4

)[

ΩA , nA]

+ . . .

+ 3πµa∑

B 6=A

G1B(x2

A) − G1B(x1

A) − G2B(x2

A) + G2B(x1

A)

f1B ,

(8.2)

where terms of O[(a/D)2] have been omitted. The unit vector nA gives the direction of sphere

2 relative to sphere 1 and thus describes the orientation of the dumb-bell. As in Chapter 1

we describe an apolar quantity by a vector instead of a line element purely for mathematical

simplicity, but, of course, our results must be independent of this artificial orientation assigned

to a dumb-bell. Applying the torque-free constraint determines the angular velocity, ΩA, from

which we obtain an equation for the evolution of the dumb-bell’s orientation

dnAdt

=[

ΩA , nA]

, (8.3)

= 1D+ξ sin(ωt)

[

I − nA ⊗ nA

]

∑

B 6=A

G1B(x2

A) − G1B(x1

A) − G2B(x2

A) + G2B(x1

A)

f1B . (8.4)

Finally, the translational motion of each dumb-bell is given by

dxcAdt

= 12

∑

B 6=A

G1B(x2

A) + G1B(x1

A) − G2B(x2

A) − G2B(x1

A)

f1B , (8.5)

where xcA denotes the ‘centre’ of the dumb-bell, xcA := (x1A + x2

A)/2. As expected, the motion

described by Eqs. (8.4) and (8.5) arises solely through interactions with other dumb-bells.

We assume that the dumb-bells are in a dilute suspension so that the separation, r, of any

given dumb-bell from its nearest neighbour may be assumed to be large compared to its size,

D. Under such circumstances the interactions between the dumb-bells may be expanded in a

power series in (D/r) and only the leading contributions retained. Integrating over a complete

cycle leads to expressions for the changes in position and orientation of the dumb-bells after a

single swimming stroke:

∆nA =∑

B 6=A

15aA32 r5BA

(


)

3(nA · rBA) + 6(nA · nB)(nB · rBA)

+ 6(nA · nB)2(nA · rBA) − 7(nA · rBA)3 − 21(nA · rBA)(nB · rBA)2

− 42(nA · nB)(nA · rBA)2(nB · rBA) + 63(nA · rBA)3(nB · rBA)2

,

(8.6)

8.2 Two dumb-bells 116

Figure 8.1: Orbits of a pair of coplanar dumb-bells. The two dumb-bells are initially parallelwith A at the origin and B placed at the corresponding point in the diagram. Several regimesof long time behaviour are found. In region P the two dumb-bells adopt a stable perpendicularconfiguration as exemplified by (i). This type of behaviour is also found in the shaded regions,however, before acquiring the perpendicular configuration the two dumb-bells tumble for aperiod of time, as shown in (ii). In the regions labelled O the dumb-bells move along paralleltrajectories with an oscillatory motion, as illustrated in (iii). Finally, along each of the dashedlines there is no rotational interaction and the dumb-bells undergo a pure translation alongstraight parallel paths.

∆xcA =∑

B 6=A

9aA32 r4BA

rBA

[

1 + 2(nA · nB)2 − 5(nA · rBA)2 − 5(nB · rBA)2

− 20(nA · nB)(nA · rBA)(nB · rBA) + 35(nA · rBA)2(nB · rBA)2]

+ 2nA

[


,

(8.7)

where rBA is the position vector of B relative to A. The amplitude A is given by

A = πD2ξAξB sin(ηBA) + π8 ξ

2Aξ

2B sin(2ηBA) , (8.8)

where ηBA is the phase of B’s swimming stroke relative to A’s. The long time behaviour of a

group of dumb-bells may be determined by numerically iterating Eqs. (8.6) and (8.7) to find

the new positions and orientations of all the dumb-bells after each swimming cycle.

8.2 Two dumb-bells

We first consider two dumb-bells lying in the yz−plane and both oriented along the z−direction.

Dumb-bell A is initially at the origin and dumb-bell B is placed on a circle of radius 4D

8.3 Linear chains of dumb-bells 117

centred on the origin. For two dumb-bells, varying the relative phase does not lead to any

qualitative changes in behaviour and therefore we consider only the case ηBA = π/2. The sole

free parameter is the angle, θ, that the position vector of B makes with the y−direction. As

this angle is varied the hydrodynamic interactions between the dumb-bells change leading to

different long time behaviour, which we illustrate in Fig. 8.1.

The predominant behaviour is for the two dumb-bells to adopt a perpendicular configuration

in which one dumb-bell is oriented parallel and the other perpendicular to their relative position

vector. This stable arrangement appears at long times for all initial configurations in the regions

labelled P in Fig. 8.1. An exemplary time series showing how the perpendicular configuration

is reached is shown in Fig. 8.1(i). The stability of this state may be seen from a linear stability

analysis of Eq. (8.6).

Similarly, linear stability analysis reveals that the rotational fixed point at θ = 0 is unsta-

ble. However, small deviations away from this fixed point do not lead smoothly to the stable

perpendicular configuration. Instead there is an initial period during which the dumb-bells

tumble, often several times, before they finally settle down, as illustrated in Fig. 8.1(ii). Ex-

actly at the fixed point, and in the absence of any fluctuations, the dumb-bells undergo a pure

translational motion, swimming cooperatively in the direction of the dumb-bell with positive

relative phase.

The fully aligned configuration with θ = π/2 is also a rotational fixed point and again,

exactly at this angle, the motion is purely translational with both dumb-bells moving in the

same direction. However, in this case, the fixed point is a centre and small deviations away

from it lead to an oscillatory cooperative motion, an exemplary time series of which is shown

in Fig. 8.1(iii). This oscillatory motion occurs throughout the regions labelled O in Fig. 8.1

and is separated from the P regions by an additional rotational fixed point at θ ≈ 65o.

8.3 Linear chains of dumb-bells

We now present examples of the cooperative effects of the hydrodynamic interaction between

many apolar swimmers, for first regular, and then random, distributions of swimmers. Consider

a chain of N identical dumb-bells all oriented along the z−direction and initially positioned at

equally spaced intervals of 5D along the y−axis. To optimise the hydrodynamic interactions

between nearest neighbours the relative phase between any two neighbouring dumb-bells is set

8.3 Linear chains of dumb-bells 118

Figure 8.2: (a) Configuration of a linear chain of dumb-bells and a schematic representationof its evolution. The bulk of the chain moves faster than the boundary, leading to dumb-bellsbeing left behind at the trailing boundary and the formation of ‘fast pairs’ at the leadingboundary. (b) The distances moved by each dumb-bell in the chain after 100, 000 swimmingstrokes for chains with different numbers of dumb-bells.

to π/2, increasing in the positive y−direction. Although this is a highly artificial configuration,

the lack of any rotation greatly simplifies the dynamics, allowing for the effect of changing N

to be clearly quantified.

We show in Fig. 8.2 the initial configuration of the chain and the distances moved by each

of the dumb-bells as their total number increases. Two features are particularly noteworthy:

firstly the behaviour for N > 2 is significantly different from that for N = 2, and secondly the

general behaviour for N > 20 shows only minor variations, indicating that an asymptotic limit

is being approached.

The evolution of the pattern of swimmers may be understood by noting that there is a

fundamental distinction between dumb-bells with two nearest neighbours and those with only

one. The former constitute what we shall call the bulk of the chain, while the latter form the

boundary. A dumb-bell in the bulk not only gets pulled along by its neighbour in front of it, but

is also pushed by its neighbour from behind. These two interactions add constructively leading

to a member of the bulk moving faster than a single isolated pair. By contrast a dumb-bell

which is on the boundary only has one nearest neighbour and hence does not benefit from this

added boost. Thus the bulk moves faster than the boundary.

This has the effect of introducing an asymmetry between the two boundaries; the dumb-

bell on the trailing boundary gets left behind while that on the leading boundary is caught

up. From Eq. (8.7) we see that the strength of interactions depends on the separation of the

dumb-bells as r−4, so that as the trailing dumb-bell gets left behind, the pull it receives from

the bulk rapidly diminishes until it becomes isolated and can no longer move. At the same time

8.4 Lattice pumps 119

the distance between the leading dumb-bell and the bulk decreases yielding a sharp increase in

the strength of interaction between itself and its neighbour. As a result the leading pair speed

up significantly and are ejected from the front of the chain.

8.4 Lattice pumps

The cooperative motion of one dimensional chains carries over to regular arrays in two dimen-

sions. As an example, consider a square lattice of dumb-bells all oriented in the z−direction and

with (x, y) positions (jL, kL), where L is the lattice constant and j, k are integers. Co-operative

directed motion can be induced by defining the phase of each dumb-bell to be φj,k = π(j+k)/2.

The entire lattice then moves uniformly along the [110] direction.

For a system of real swimmers this state could not be sustained; we have found that it is

unstable to any imperfections of the lattice such as boundaries or fluctuations in the position

of the swimmers. This instability of long range coherent states is in agreement with analytic

work by Ramaswamy and co-workers [98, 99, 108] and numerical simulations of Saintillan and

Shelly [113]. However, one might envisage tethering fabricated dipolar swimmers to a substrate

and aligning and activating them with a magnetic field. Such a set-up would act as a micon-

scale pump. We estimate v ∼ µms−1, which is similar to the velocities achieved by bacteria.

It should be noted that the interactions scale with separation as r−4, so that a small decrease

in lattice spacing will provide a substantial increase in flow speed.

8.5 Suspensions of dumb-bells

We next consider a suspension of N dumb-bells initially dispersed throughout a cubic box of

side L = 20D with random positions, orientations and relative phases. To avoid singularities

in the hydrodynamic interactions a short distance cut-off is employed when the separation

between any two dumb-bells becomes less than 0.5D. At separations of order D an expansion

of the Oseen tensor in powers of D/r does not converge rapidly. To overcome these difficulties

we instead use an expansion in powers of ξ/r to better describe the near field hydrodynamics.

The combination of a random initial configuration and apolar symmetry means that there is

no prefered direction for the motion, and on average the velocity is zero. In continuum models of

apolar active fluids spontaneous symmetry breaking can lead to a state with non-zero average

velocity [175, 176], however we have not observed any such transitions in our simulations of

8.5 Suspensions of dumb-bells 120

Figure 8.3: Collective properties of a suspension of interacting (I) dumb-bell swimmers (apolar)and (II) three-sphere swimmers (polar). Left: mean speed of the swimmers as a function oftheir number density. The line indicates a linear fit. Centre: mean square displacement asa function of time for number densities 0.1 (×) and 0.02 (⋄). Right: diffusion constant forincreasing number of swimmers.

dumb-bells. In the absence of a net velocity, the mean speed provides a measure of the degree

of collective activity. Fig. 8.3(I) shows that the mean speed of the dumb-bells increases linearly

with the number density n = N/L3. This agrees with a simple scaling argument: since every

dumb-bell will interact with every other one, the total number of interactions scales as n2.

Balancing this against the rate of dissipation of energy predicts that the mean speed should

increase linearly with n. The same linear scaling, but tending to a finite value as N → 1

because of the finite speed of a single swimmer, is also found for a suspension of polar swimmers

(Fig. 8.3(II)).

A measure of the nature of the collective motion generated by the interacting dumb-bells

is the mean square displacement, 〈∆r2(t)〉e = 〈|r(t) − r(0)|2〉e, where 〈·〉e denotes an ensemble

average. This is plotted in Fig. 8.3 for number densities of n = 0.02 and n = 0.1. At long times,

and for large number densities, a scaling form 〈∆r2(t)〉e ∼ tz develops, with an exponent of

z = 1 indicating that the suspension of dumb-bells is behaving diffusively. At smaller number

densities the behaviour is more sporadic because the large average separation between dumb-

bells greatly reduces the strength of the interactions between them. The collective motion is

then dominated by those fluctuations in the local number density which bring two dumb-bells

close enough to allow them to move appreciably.

The mean square displacement of a suspension of dumb-bells shows qualitatively different

8.6 Discussion 121

behaviour to that of a suspension of polar swimmers. In the latter case, shown in Fig. 8.3(II),

the mean square displacement is ballistic at short times (z = 2), with a cross-over to diffusion

(z = 1) at longer times. Moreover, since this cross-over is due to the randomisation of swimmer

orientations through hydrodynamic interactions, it occurs at later and later times as the number

density is reduced. For polar swimmers this leads to a diffusion constant which decreases as

the number density is increased. For apolar swimmers the opposite is true; their motion arises

solely from hydrodynamic interactions and the diffusion constant increases with increasing

number density.

8.6 Discussion

The aim of this Chapter has been to discuss the motion of a simple model of apolar swimmers,

systems which move at zero Reynolds number only in the presence of other swimmers. We

have demonstrated that, as is the case for colloids [179] and polar swimmmers [165, 166],

hydrodynamic interactions lead to complex collective behaviour.

Particular observations are that, for two dumb-bell swimmers which are initially parallel,

the most likely final state corresponds to the axes of the swimmers lying at right angles.

Oscillatory trajectories are also observed. For regular arrays of swimmers cooperation between

hydrodynamic pair interactions can lead to simple flow fields. In particular, a square array of

dumb-bells produces a constant flow and hence, if the dumb-bells were fixed in position, could

act as a pump. For a large number of dumb-bell swimmers, initialised with random positions

and phases, the mean speed is linear in the number of dumb-bells, reflecting the energy pumped

into the system by the hydrodynamic interactions. The mean square displacement evolves

linearly in time showing the expected diffusive behaviour.

There is much further work to be done to explore the phase space of apolar swimmers, for

example considering initial configurations for which pairs move out of the plane, three dumb-

bell orbits, and the role of the distribution of phases on multi-dumb-bell motion. For more

than two dumb-bells the relative phase becomes an important variable; since the pairwise

interactions between three or more dumb-bells cannot simultaneously take their maximum

value the system exhibits a type of frustration. It is also interesting to consider the effect of

moving away from the zero Reynolds number limit [169], and to ask which of the properties

of dumb-bells provide a generic representation of the class of apolar swimmers. Comparing

8.6 Discussion 122

simple microscopic models of polar and apolar swimming may help to formulate the correct

continuum theory of swimmers, and to link the microscopic and continuum length scales.

Chapter 9

Conclusion

“What distressingly few of us realise is that almost

all of these Taylors, except that of the Taylor series,

are the same Taylor, Geoffrey Ingram.”

The end of this thesis by no means marks the end of either of the two topics we have

described. We take the opportunity to summarise our work and indicate some topics for future

research.

Blue phases

Blue phases have seen a resurgence of interest in the past decade, prompted by the development

of new experimental techniques to increase their stability and to exploit their unique properties

for device applications. A thorough understanding of their response to an applied electric field

is therefore important, since most liquid crystal devices rely upon the action of an electric field

for their operation. Although this is a classic problem, worked on extensively in the 1980s,

the agreement between theory and experiment has always been incomplete due to the inherent

difficulty of the analytic calculations. In Chapter 3 we were able to show that the agreement

may be improved, at least in part, by a numerical approach, which has the added benefit of

describing both the static changes induced in the blue phase texture and the dynamics of their

response.

Using a simple numerical technique for determining the redshift (§ 2.4) we were able to

simulate the electrostriction of the cubic blue phases, obtaining for the first time both the

correct order of magnitude of the components of the electrostriction tensor and the anomalous

123

Conclusion 124

electrostriction of blue phase I (§ 3.1). The numerical approach allows for an unprecedented

description of the dynamics of electric field effects, allowing the pathway for textural transitions

and complete unwinding of the blue phase to be determined. We were able to reproduce the

blue phase I-blue phase X transition (§ 3.3) leading to the first determination of the detailed

structure of the latter texture.

A full determination of the sequence of textural transitions that occur under the applica-

tion of an electric field remains an important goal. Although a partial analytic calculation

was performed by Hornreich et al in 1990 [79], the calculations were too extensive to allow

blue phase X to be incorporated. The known differences between analytically and numerically

calculated phase diagrams in zero field suggests that there may be similar discrepancies here.

Furthermore, there is the interesting possibility that the three-dimensional hexagonal texture

described by Hornreich et al [79] and included in their calculations may not be the correct

phase. This is because they took their three-dimensional texture to be generated from the

fundamental wavevectors for the two-dimensional hexagonal texture together with higher har-

monics. However, there is also the possibility that the three-dimensional texture is generated

by sub-harmonics, exactly as for the O2 texture corresponding to blue phase II.

It would also be interesting to investigate experimentally the flexoelectric response of the

blue phases [49]. Flexoelectric effects in the cholesteric phase offer a number of benefits for

potential device applications, for example faster switching times and a response that is linear in

the field strength for small fields, so they may afford similar benefits in blue phase devices. In

Chapter 4 we identified three effects that might be expected for the blue phases: a change in the

size and shape of the unit cell, or electrostriction, quadratic in the field strength, a distortion

of the double twist cylinders at small fields, linear in the field strength, and a series of textural

transitons to tetragonal and finally two-dimensional hexagonal textures. It is unfortunate that

most of these effects are similar to those observed under dielectric coupling; the electrostriction

is also quadratic in field strength and the textural transitions lead to structures with the

same symmetries. This will undoubtedly make it difficult to distinguish between effects of

dielectric and flexoelectric origin experimentally. The clearest indication that a phenomenon

is flexoelectric in origin is if the response is linear in the field strength, as in the rotation of the

optic axis in the flexoelectro-optic effect. The distortion of double twist cylinders in the blue

phases represents one such linear response, which it may be possible to observe directly using

confocal microscopy [84,85].

Conclusion 125

The origin of the stability of the wide temperature range blue phase discovered by Coles

and Pivnenko [50] is still not understood. They have proposed that it might be flexoelectric

in origin, based on the observation that the same bimesogenic molecules that yield the stable

blue phase also exhibit large flexoelectric couplings. The status of polarisation fields produced

internally in the cubic blue phases, and their consequences for energetic stability, remain to be

determined.

Finally, a new direction that we are currently investigating is the influence of colloidal

particles on the properties and stability of the blue phases. When a colloidal particle is added

to a nematic liquid crystal, the anchoring conditions on the surface of the colloid are usually

such that it is accompanied by the creation of disclination lines and a concomitant increase in

the free energy. Not so for a blue phase, where the colloidal particles can attach themselves

to the pre-existing disclination lines, occupying part of their position within the unit cell, and

thus removing a length of disclination line. Since the disclination lines are energetically costly,

if the energy associated with the surface anchoring of the colloid is not too large, this will lead

to an overall reduction in the free energy, thereby aiding stabilisation of the blue phases.

Swimmers

Simple models, such as the Najafi-Golestanian swimmer, capture the essential feature of mi-

croorganisms in that they are able to swim by changing their shape. The usefulness of these

models is that they remain analytically and numerically tractable without overly simplifying

the details of their means of propulsion. The dependence of the hydrodynamic interactions on

the properties of the swimming stroke can then be correctly determined and used as a guide

to constructing more coarse-grained theories for the description of the collective behaviour of

suspensions of swimmers.

We have shown in Chapter 6 that at a detailed, interswimmer level these interactions are

not well described by the dipolar form suggested by simple scaling arguments and assumed in

some previous studies [101,108,110]. Instead the interactions are higher order, decaying as r−4BA

in the far field, and exhibit a sensitive dependence on the relative phase of the two swimmers.

This sensitivity to the relative phase is important and allows for a fundamental distinction to

be made between active and passive interactions.

An important next step is to extend the approach we developed in Chapter 6 to other

simple models to see how substantially the details of the interactions vary between different

Conclusion 126

types of swimmers. An obvious extension is to linked sphere swimmers with transverse, instead

of longitudinal, swimming strokes. A substantial part of the calculation for this case has

already been performed, with a number of noteworthy features arising from the results. The

most important is simply geometrical. Two vectors are required to describe the motion of a

transverse swimmer: a vector n giving the direction of its long axis and a vector m giving the

direction perpendicular to the long axis along which the oscillations take place.

Since, for such a swimmer, the forces act perpendicular to the long axis they will, in general,

produce a torque that must be balanced by a bodily rotation so that the overall motion of the

swimmer imparts no net torque on the surrounding fluid. We have found, however, that this

bodily rotation can be eliminated by a suitable choice of the parameters of the swimming

stroke, that choice corresponding precisely to the optimal swimming strategy. For this special

case, the interactions between the swimmers both simplify greatly and exhibit some unexpected

cancellations, leading to an active rotation that decays with separation as r−5BA and an active

advection that decays even faster, as r−6BA.

The dependence of these active interactions on the relative phase is also interesting, ap-

pearing in both the advection and the rotation through a factor sin(ηBA). Thus it is possible

to eliminate the active interactions entirely by arranging for all the swimmers in a suspension

to be exactly in phase, an option not available to Najafi-Golestanian swimmers since their ac-

tive interactions vary cosinusoidally with the relative phase between them. Understanding the

consequences this has for the collective behaviour of groups of the different types of swimmers

will be an important future step.

Already it is clear that the nature of the swimming stroke of an individual can have an

appreciable effect on the form of the detailed interactions between them. However, some

features of swimmer-swimmer interactions are generic. For example, the scattering phenomena

we described in Chapter 7 apply equally well to swimmers with transverse strokes, with the same

types of scattering events, turn and exchange, and the exact recovery of the initial state, at least

for planar scattering geometries. Indeed, our argument in Chapter 7 was that the phenomenon

was due to symmetry, a consequence of kinematic reversibility, and therefore applicable to any

swimmer, since, in principle, a T-dual can always either be found or fabricated.

Finally, we should like to be able to say more about the large scale behaviour of suspensions

of swimmers and the appropriate form for a continuum description. Our analysis of apolar

dumb-bells in Chapter 8 represents a start to this, but it is little more than a first step. Work

Conclusion 127

is currently under way to extend these results and provide a more thorough comparison between

the collective properties of polar and apolar swimmers. Our aim will be to make a connection

to prominent features of continuum theories such as orientational ordering, spontaneous flow

and anomalous density fluctuations.

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